Probability for the following events will be :-
The probability of an event E,P(E)=
Number of favourable outcomes / Total number of outcomes
= n(E) / n(S)
Applying the concept of probability as shown above, let's calculate all the probabilities one by one.
(i) Probability of getting a king of red colour:
We know that, there are 26 red cards, 13 each of hearts and diamonds. Hence, there will be 1 king each in hearts and diamonds.
∴ Number of kings of red colour, n(E)=2
Also, total number of cards, n(S)=52
∴ Probability of getting a king of red colour
= n(E) / n(S)
= 2 / 52
⇒ 1 / 26
Hence, the required probability is 1 / 26
(ii) Probability of getting a face card:
We know that, each suit has 3 face cards (Jack, Queen and King). Hence, there will be 12 face cards in total.
∴ Number of face cards, n(E)=12
Also, total number of cards, n(S)=52
∴ Probability of getting a face card
= n(E) / n(S)
= 12 / 52
⇒ 3 / 13
Hence, the required probability is 3 / 13
(iii) Probability of getting the jack of hearts:
We know that, there is only one jack in hearts.
∴ Number of jack of hearts, n(E)=1
Also, total number of cards, n(S)=52
∴ Probability of getting the jack of hearts =
n(E) / n(S)= 1 / 52
Hence, the required probability is 1 / 52
(iv) Probability of getting a red face card:
We know that, there are 26 red cards, 13 each of hearts and diamonds. Both hearts and diamonds have 3 face cards each.
∴ Number of red face cards, n(E)=6
Also, total number of cards, n(S)=52
∴ Probability of getting a red face card =
n(E) / n(S) = 6 / 52
⇒ 3 / 26
Hence, the required probability is 3 / 26
(v) Probability of getting a spade:
We know that, there are 13 cards of spades.
∴ Number of spades, n(E)=13
Also, total number of cards, n(S)=52
∴ Probability of getting a spade =
n(E) / n(S) = 13 / 52
⇒ 1 / 4
Hence, the required probability is 1 / 4
(vi) Probability of getting the queen of diamonds:
We know that, there are 13 cards of diamonds. There is 1 queen of diamonds.
∴ Number of queens in diamonds, n(E)=1
Also, total number of cards, n(S)=52
∴ Probability of getting the queen of diamonds =
n(E) / n(S) = 1 / 52
Hence, the required probability is 1/52
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for the range c6:c41, create a new conditional format that changes the font color to red for values that are greater than or equal to 50.
The following steps would apply conditional formatting to the range:
Select the range of cells C6 : C41Click Conditional Formatting on the Home tabSet the conditionPoint to Color Scales, and then select the font color as red colorHow to apply the conditional formatting?The given parameters in the question are:
Range = C6 : C41Condition = cell that are greater than or equal to 50.Action = font color as red colorThe Conditional Formatting is located on the home tab
So, the first step to select the range and locate the Conditional Formatting option
Then set the necessary conditions and formats
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URGENT PLEASE ANSWER
Answer:
y = 371.17×1.15^x1135 in month 8Step-by-step explanation:
You want an exponential regression function for the data in the table.
a. Exponential regressionA regression function of most any kind can be found using a graphing calculator or spreadsheet. The attachment shows the function is approximately ...
y = 371.17 × 1.15^x
where y is the number sold, and x is the month.
b. Number soldUsing x=8, the predicted number sold for month 8 is ...
y = 371.17 × 1.15^8 ≈ 1135.4175 ≈ 1135
Sales for month 8 will be about 1135.
How do type an exponent on a chromebook laptop?
Step-by-step explanation:
Press ''CTRL + /'' to access the list of features.
Find the “Text Formatting” section.
Choose “Superscript” from the list of options.
On the right-hand side, you'll see the ''CTRL +. '' shortcut. Use it to make the number or letter in your text appear in exponent form.
Step-by-step explanation:
1. Place your cursor where you want an exponent. For example, if you want to place an exponent after the number 10 in a document, place your cursor directly after the 10 with no space.
2. Type Alt+0185 for the exponent 1. For example, you can type the number 10¹ holding the Alt key and typing 0185.
3. Type Alt+0178 for the exponent 2. For example, you can type the number 10² holding the Alt key and typing 0178.
4. Type Alt+0179 for the exponent 3. For example, you can type the number 10³ by holding the Alt key and typing 0179.
Rather than being irrelevant or unhelpful, residuals in a simple linear regression model can be quite informative. Which of the following, in your opinion, residuals might indicate about the underlying population and the model?
Regression analysis will indicate about the underlying population and the model.
It is one of the most used and most powerful multivariate statistical techniques for it infers the existence and form of a functional relationship in a population. Once you learn how to use regression, you will be able to estimate the parameters { the slope and intercept } of the function that links two or more variables. With that estimated function, we will be able to infer or forecast things like unit costs, interest rates, or sales over a wide range of conditions.
Though the simplest regression techniques seem limited in their applications, statisticians have developed a number of variations on regression that greatly expand the usefulness of the technique. And again, the t-distribution and F-distribution will be used to test hypotheses.
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Assuming the two clocks start synchronized, how much do sidereal and solar clocks differ after one day?
a. about 4 minutes
b. about an hour
c. about two hours
d. 24 hours
e. none of the above
If we assume the two clocks start synchronized , then the sidereal and solar clocks differ after one day , (a) about 4 minute .
What is Solar Day ?
The rotation period of the Earth with respect to the Sun, is called the solar day .
What is Sidereal Day ?
The sidereal day can be defined in terms of the rotation period of the Earth with respect to stars.
To complete a solar day, the Earth needs to rotate an additional amount, that is equal to 1/365 of a full turn.
So , the time required for this extra rotation will be 1/365 of a day, or about 4 minutes.
Hence , the solar day is about 4 minutes longer than the sidereal day.
Therefore , the correct option is (a) about 4 minutes .
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Problem 3: Integration via the Substitution Method ( 16 points) Evaluate each of the following. Explain and show your work. You do not need to simplify your final answer. a)
(8
points
)∫ 0
3
ㅌㅣ
(2e 5t
+3) 2
3e 5t
dt
VALUE: b) (8 points)
∫( sin 2
(t 2
)+1
tcos(t 2
)
)dt
The value of the given integral [tex]\int^{\text{In}3/5}_{0}\frac{3e^{5t}}{(2e^{5t}+3)^2}dt[/tex] is 2/75.
The given integral is [tex]\int^{\text{In}3/5}_{0}\frac{3e^{5t}}{(2e^{5t}+3)^2}dt[/tex].
We have to evaluate the given integral.
The given integral is definite integral. To evaluate the given integral we firstly solve the integral then we put the value in the solved integral to find the value.
We solving the integral [tex]\int\frac{3e^{5t}}{(2e^{5t}+3)^2}dt[/tex]
Let u = 2e^{5t}+3
Now differentiating
du/dt = 10e^{5t}
e^{5t}dt = du/10
[tex]\int\frac{3e^{5t}}{(2e^{5t}+3)^2}dt=\frac{3}{10}\int\frac{1}{u^2}dt[/tex]
[tex]\int\frac{3e^{5t}}{(2e^{5t}+3)^2}dt=\frac{3}{10}\int{u^{-2}}dt[/tex]
[tex]\int\frac{3e^{5t}}{(2e^{5t}+3)^2}dt[/tex] = 3/10 [u^{-2+1}/(-2+1)]+C
[tex]\int\frac{3e^{5t}}{(2e^{5t}+3)^2}dt[/tex] = 3/10 [u^{-1}/(-1)]+C
[tex]\int\frac{3e^{5t}}{(2e^{5t}+3)^2}dt[/tex] = -3/10(1/u)+C
[tex]\int\frac{3e^{5t}}{(2e^{5t}+3)^2}dt[/tex] = -3/10(1/2e^{5t}+3)+C
Now;
[tex]\int^{\text{In}3/5}_{0}\frac{3e^{5t}}{(2e^{5t}+3)^2}dt=-\frac{3}{10}\left[\frac{1}{2e^{5t}+3}\right]^{\text{In}3/5}_{0}[/tex]
[tex]\int^{\text{In}3/5}_{0}\frac{3e^{5t}}{(2e^{5t}+3)^2}dt=-\frac{3}{10}\left[\frac{1}{2e^{5({\text{In}3/5})}+3}-\frac{1}{2e^{5(0)}+3}\right][/tex]
[tex]\int^{\text{In}3/5}_{0}\frac{3e^{5t}}{(2e^{5t}+3)^2}dt=-\frac{3}{10}\left[\frac{1}{2e^{({\text{In}3})}+3}-\frac{1}{2e^{0}+3}\right][/tex]
[tex]\int^{\text{In}3/5}_{0}\frac{3e^{5t}}{(2e^{5t}+3)^2}dt=-\frac{3}{10}\left[\frac{1}{2\cdot3+3}-\frac{1}{2+3}\right][/tex]
[tex]\int^{\text{In}3/5}_{0}\frac{3e^{5t}}{(2e^{5t}+3)^2}dt=-\frac{3}{10}\left[\frac{1}{6+3}-\frac{1}{2+3}\right][/tex]
[tex]\int^{\text{In}3/5}_{0}\frac{3e^{5t}}{(2e^{5t}+3)^2}dt[/tex] = -3/10[1/9 - 1/5]
[tex]\int^{\text{In}3/5}_{0}\frac{3e^{5t}}{(2e^{5t}+3)^2}dt[/tex] = -3/10[5-9/45]
[tex]\int^{\text{In}3/5}_{0}\frac{3e^{5t}}{(2e^{5t}+3)^2}dt[/tex] = -3/10[-4/45]
[tex]\int^{\text{In}3/5}_{0}\frac{3e^{5t}}{(2e^{5t}+3)^2}dt[/tex] = 2/75
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The right question is:
Problem 3: Integration via the Substitution Method ( 16 points) Evaluate each of the following. Explain and show your work. You do not need to simplify your final answer.
[tex]\int^{\text{In}3/5}_{0}\frac{3e^{5t}}{(2e^{5t}+3)^2}dt[/tex]
Quadrilateral ABCD is translated to get quadrilateral A′B′C′D′. Vertex A is at (-5, 2), and vertex A′ is at (2, -2).
Quadrilateral ABCD is translated
. If vertex B is at (-6, 5), then vertex B′ is at
The vertex B' is at (1, 1)
How to find the calculation?Let's update the guidelines for translating a point.
The image of the point (x, y) is (x + h, y) T (x, y) (x + h, y) if the point (x, y) is translated horizontally to the right by h units.The image of the point (x, y) is (x - h, y) T (x, y) if the point (x, y) is translated horizontally to the left by h units (x - h, y)The image of the point (x, y) is (x, y + k) (x + h, y) T (x, y) (x, y + k) if the point (x, y) is translated vertically up by k units.The image of the point (x, y) is (x, y - k) T (x, y) if the point (x, y) is translated down vertically by k units (x, y - k)Let's apply the guidelines to resolve our issue.
Vertex A equals (-5, 2)
Vertex A' equals (2, -2)
The two points' coordinates changed, indicating that the
horizontal and vertical translations of a quadrilateral
A's x-coordinate is equal to -5.
A' has an x-coordinate of 2
Which indicates it goes to the right, therefore adhere to the first guideline presented above.
The translation formula is T (x, y) (x + h, y).
∴ x → x + h
x = -5, and x + h = 2.
∴ -5 + h = 2
Increase both sides by 5.
∴ -5 + 5 + h = 2 + 5
∴ h = 7
∴ The quadrilateral is moved seven units to the right.
A has a y-coordinate of 2
A"s y-coordinate is equal to -2.
Which means it descends, therefore apply the fourth rule listed above.
T (x, y) is the translational rule (x, y - k)
∴ y → y - k
Because y = 2 and y - k = -2,
∴ 2 - k = -2
Increase K on both sides.
∴ 2 - k + k = -2 + k
∴ 2 = -2 + k
Increase both sides by 2.
∴ 2 + 2 = -2 + 2 + k
∴ 4 = k
The quadrilateral is down four units.
Let's locate B.
T (x, y) (x + h, y - k) is the translational rule.
∵ B = (-6, 5) (-6, 5)
∵ h = 7 and k = 4
∴ B' = (-6 + 7, 5 - 4)
∴ B' = (1, 1) (1, 1)
Vertex B is located at (1, 1).
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two different two-digit whole numbers are selected at random. what is the probability that their product is less than 200. express your answer as a common fraction. (hints: (l) there are 90 different two-digit numbers, (2) the pair {10, 11} produces the smallest product and the pair {11, 18} produces the largest product less than 200).
The probability that the product of the two two-digit numbers is less than 200 is given as follows:
43/8010.
How to calculate the probability?A probability is calculated as the division of the number of desired outcomes in the context of the experiment by the number of total outcomes.
There are 90 different two-digit numbers, hence the number of total outcomes for the product is of:
90 x 89 = 8010.
(the numbers have to be different)
The desired outcomes which result in a product of less than 200 are of given as follows:
10 multiplied by 9 numbers, from 11 to 19.11 multiplied by 10, 12, 13, 14, 15, 16, 17, 18. (8 numbers).12 multiplied by 10, 11, 13, 14, 15, 16. (6 numbers).13 multiplied by 10, 11, 12, 14, 15 (5 numbers).14 multiplied by 10, 11, 12, 13. (4 numbers).15 multiplied by 10, 11, 12, 13. (4 numbers).16 multiplied by 10, 11, 12. (3 numbers).17 multiplied by 10, 11. (2 numbers).18 multiplied by 10, 11. (2 numbers).Hence the number of desired outcomes is given as follows:
9 + 8 + 6 + 5 + 4 + 4 + 3 + 2 + 2 = 43.
Meaning that the probability is of:
43/8010.
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A The numbers 14 and 29 have more than one common factor
B The number 29 is odd
C The greatest common factor of 14 and 29 is 1
D The number 14 is even
Answer:
The greatest common factor of 14 and 29 is 1
Step-by-step explanation:
Hope this helped you
I NEED HELP THIS IS DUE TONIGHT
Answer:
The park has 20 climbing activities.
Step-by-step explanation:
Given,
Total activities = 50
Percent of climbing activities = 40%
Number of climbing activities = 40% of total activities
Number of climbing activities = 40/100 * 50
Number of climbing activities = 0.40 * 50 = 20
Hope this helps!!
can someone help i will give brainlest:)
A. Width = 16.5 ft B.Width = 55.25 ft
Height = 9.3 ft Height = 91 ft
Area = 153.45 ft² Area = 5027.75 ft²
The formula for Area of rectangle:The area of the rectangle is a place covered by a rectangle in a 2D plane
The formula for Area of the rectangle is given by
Area = Length × WidthHere we have
1 in = 3 feet and 1 cm = 6.5 ft
Calculation for rectangle A :
In rectangle A
Width = 5.5 in
=> Width in feet = 5.5 × 3 feet = 16.5 ft
Height = 3.1 in
=> Height in feet = 3.1 × 3 feet = 9.3 ft
By the given formula
Area = 16.5 ft × 9.3 ft = 153.45 ft²
Calculation for rectangle B :
In rectangle B
Width = 8.5 cm
=> Width in feet = 8.5 × 6.5 feet = 55.25 ft
Height = 14 cm
=> Height in feet = 14 × 6.5 feet = 91 feet
By the given formula
Area = 55.25 ft × 91 ft = 5027.75 ft²
Therefore,
A. Width = 16.5 ft B. Width = 55.25 ft
Height = 9.3 ft Height = 91 ft
Area = 153.45 ft² Area = 5027.75 ft²
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look at the picture
The quadratic inequality (18 / 7) · x² + (54 / 7) · x - 180 / 7 < y contains all the ordered pairs outside the parabola.
What is parabola?
A parabola could be a U-shaped plane curve wherever any purpose is at an equal distance from a hard and fast purpose (known because the focus) and from a fixed line that is understood because the directrix.
Main body:
In this problem we must determine a quadratic inequality of the form:
a · x² + b · x + c < y
Where a, b, c are the real coefficients of the inequality.
First, determine the coefficients of the quadratic inequality (parabola) by solving the system of linear equations:
25 · a - 5 · b + c = 0
4 · a + 2 · b + c = 0
a + b + c = - 18
(a, b, c) = (18 / 7, 54 / 7, - 180 / 7)
Second, write the quadratic in equation:
(18 / 7) · x² + (54 / 7) · x - 180 / 7 < y
hence , the equation is (18 / 7) · x² + (54 / 7) · x - 180 / 7 < y
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y varies inversely with the square root of x. When x=64, then y=12. Find y
when x=36
The value of y is 16.
Inverse proportionality:When one quantity's value increases compared to another's decrease or vice versa, it is said that the two are inversely proportionate.
It implies that the two quantities exhibit opposing behaviors in nature. For instance, the relationship between time and speed is inverse.
Inverse proportionality will be represented by the symbol '∝'. When two values x and y are Inverse proportional to each other then
x ∝ 1/y
Here we have
y varies inversely with the square root of x
=> [tex]y\propto\frac{1}{\sqrt{x} }[/tex]
=> [tex]y\ = \frac{k}{\sqrt{x} }[/tex]
=> [tex]\sqrt{x} y = k[/tex] ----- (1)
At x = 64, and y = 12
=> √64 × 12 = k
=> k = (8) × 12
=> k = 96
Thus the proportionality constant k = 96
Now Take x = 36 and k = 96 in equation (1)
=> √36 × y = 96
=> 6 × y = 96
=> y = 16
Therefore,
The value of y is 16.
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The area of a rectangle is 35 m², and the length of the rectangle is 3 m more than double the width. What is the dimensions of the rectangle?
The dimensions of the rectangle with area 35 m² is
length = 7.6 m and width = 4.6 m
How of find the dimension of the rectangleThe dimension of the rectangle is calculated using the formula for area of rectangle
= length x width
where
length = 3 + w width = warea of rectangle
35 = (w + 3) * w
35 = w² + 3w
w² + 3w - 35 = 0
solving for w gives
w = 4.6 OR -7.6
using the positive value, the w = 4.6 m, l = 4.6 + 3 = 7.6 m
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l
Decide if the given value of x is a critical number for f, and if so, decide whether the point for x on f is a relative minimum, relative maximum, or neither.
f(x) = ( x + 5)^4; x = - 5
A. Critical number; maximum at (-5 , 0)
B. Critical number; minimum at (-5 , 0)
C. Not a critical number.
D. Critical number but not an extreme point.
The value of the function after evaluation is , f(x)=0 . Hence it is neither a minima nor a maxima.
f(x) =(x + 5)^4; x = - 5
=(-5+5)^4
=(0)^4
=0
The critical value is crucial for accurately portraying a range of properties in statistics. The critical value can be helpful for proving or disproving ideas when they are tested, in addition to validity and accuracy.
If you're taking a statistics course or are just interested in how these concepts work, it is essential to comprehend critical value and how to compute it before analyzing other statistical functions, such as margin of error and significance.
When calculating the margin of error for a set of data, statisticians utilize a measurement known as the critical value.
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Im order to save money for prom this weekend Tom is going to walk his neighbors dog for 6 per hour
The system of inequalities represents this solution is:
6c + 7.5d ≥ 75
d + c ≤ 15.
What is inequality?A statement of an order relationship, greater than or equal to, less than, less than or equal to- between two numbers or algebraic equations.
Now it is given that,
Money from walking neighbor's dog = $6 per hour
Money from car wash = $7.5 per hour
Total Number of hours of work not more than 15 hours
Now, if d represents the number of hours walking his neighbor's dog and c represents the number of hours washing car
So, money from walking neighbor's dog = 6c
and money from car wash = 7.5d
Therefore total money saved = 6c + 7.5d
∵ Total Number of hours of work is not more than 15 hours
⇒ d + c ≤ 15
∵ Tom needs to make at least $75 to cover prom expenses
⇒ 6c + 7.5d ≥ 75
The system of inequalities represents this solution is:
6c + 7.5d ≥ 75 ;
d + c ≤ 15.
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The complete question is as follows:
To save money for prom this weekend, Tom is going to walk his neighbor's dog for $6 per hour and wash cars for $7.50 per hour. His mother said he can work no more than 15 hours to ensure he keeps up with his homework. Tom needs to make at least $75 to cover prom expenses. If d represents the number of hours walking his neighbor's dog and c represents the number of hours washing cars, which system of inequalities represents this solution?
Mark all of the transformations that will result in the image changing size from the pre-image.
A Dilation of a scale factor of 0.25
B Rotation of 180 degrees.
c Reflection over the x-axis.
D Dilation of a scale factor of 3.
E Translation of 5 to the right and 3 up.
Answer:
Step-by-step explanation:
c
14. Which of these statements correctly describes the function y = 4x + 9?
A. It is not a linear function, and its graph is not a straight line.
B. It is a linear function, but its graph is not a straight line.
C. It is not a linear function, but its graph is a straight line.
D. It is a linear function, and its graph is a straight line.
Answer:
Step-by-step explanation:
because there is no change in slope. The graph will always be strait in this
how to interpret the slope? and what is the slope?
Answer:
100
Step-by-step explanation:
0 20 40 60 100
5. l
10. l
15. l
20. l
√100 +4, √81 + A, √64 + 6, √49 + B, √36 +8, √25+ C
Answer:
Step-by-step explanation:
Simplifying square roots
Example
Let's simplify \sqrt{75}
75
square root of, 75, end square root by removing all perfect squares from inside the square root.
We start by factoring 757575, looking for a perfect square:
75=5\times5\times3=\blueD{5^2}\times375=5×5×3=5
2
×375, equals, 5, times, 5, times, 3, equals, start color #11accd, 5, squared, end color #11accd, times, 3.
We found one! This allows us to simplify the radical:
\begin{aligned} \sqrt{75}&=\sqrt{\blueD{5^2}\cdot3} \\\\ &=\sqrt{\blueD{5^2}} \cdot \sqrt{{3}} \\\\ &=5\cdot \sqrt{3} \end{aligned}
75
=
5
2
⋅3
=
5
2
⋅
3
=5⋅
3
So \sqrt{75}=5\sqrt{3}
75
=5
3
square root of, 75, end square root, equals, 5, square root of, 3, end square root.
Want another example like this? Check out this video.
Practice
PROBLEM 1.1
Simplify.
Remove all perfect squares from inside the square root.
{\sqrt[]{12}}=
12
=root, start index, end index, equals
Explain
Want to try more problems like these? Check out this exercise.
Simplifying square roots with variables
Example
Let's simplify \sqrt{54x^7}
54x
7
square root of, 54, x, start superscript, 7, end superscript, end square root by removing all perfect squares from inside the square root.
First, we factor 545454:
54=3\cdot 3\cdot 3\cdot 2=3^2\cdot 654=3⋅3⋅3⋅2=3
2
⋅654, equals, 3, dot, 3, dot, 3, dot, 2, equals, 3, squared, dot, 6
Then, we find the greatest perfect square in x^7x
7
x, start superscript, 7, end superscript:
x^7=\left(x^3\right)^2\cdot xx
7
=(x
3
)
2
⋅xx, start superscript, 7, end superscript, equals, left parenthesis, x, cubed, right parenthesis, squared, dot, x
And now we can simplify:
\begin{aligned} \sqrt{54x^7}&=\sqrt{3^2\cdot 6\cdot\left(x^3\right)^2\cdot x} \\\\ &=\sqrt{3^2}\cdot \sqrt6 \cdot\sqrt{\left(x^3\right)^2}\cdot \sqrt x \\\\ &=3\cdot\sqrt6\cdot x^3\cdot\sqrt x \\\\ &=3x^3\sqrt{6x} \end{aligned}
54x
7
=
3
2
⋅6⋅(x
3
)
2
⋅x
=
3
2
⋅
6
⋅
(x
3
)
2
⋅
x
=3⋅
6
⋅x
3
⋅
x
=3x
3
6x
Practice
PROBLEM 2.1
Simplify.
Remove all perfect squares from inside the square root.
\sqrt{20x^8}=
20x
8
=square root of, 20, x, start superscript, 8, end superscript, end square root, equals
Explain
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More challenging square root expressions
PROBLEM 3.1
Simplify.
Combine like terms and remove all perfect squares from inside the square roots.
2\sqrt{7x}\cdot 3\sqrt{14x^2}=2
7x
⋅3
14x
2
=2, square root of, 7, x, end square root, dot, 3, square root of, 14, x, squared, end square root, equals
Explain
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A side of the equilateral triangle 'A' is twice the length of a side of
another equilateral triangle 'B'. How many triangles of 'B' will fit
into triangle 'A'?​
Total four triangles of B fit into triangle A as length of the equilateral triangle A is twice the length of triangle B.
As given in the question,
Types of triangles : Equilateral triangle
Number of triangle = 2
Triangle A and Triangle B
Let 'x' be the length of the triangle A
And 'y' be the length of the triangle B
As per given condition:
x = 2y
Area of equilateral triangle A = (√3/4)x²
Area of equilateral triangle B = (√3/4)y²
Now x = 2y
⇒Area of equilateral triangle A = (√3/4)(2y)²
⇒Area of equilateral triangle A = (√3/4)4y²
⇒Area of equilateral triangle A = 4 [(√3/4)y²]
⇒Area of equilateral triangle A = 4 ( area of equilateral triangle B)
Therefore, total number of four equilateral triangle of B fit into equilateral triangle A.
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Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. (If an answer does not exist, enter DNE. Round your answers to one decimal place. Below, enter your answers so that ∠B1 is larger than ∠B2.)
a = 36, c = 44, ∠A = 31°
Possible triangles that satisfy the given conditions are:
Triangle 1: a = 36, b = 43.5, c = 44, A = 31, B = 63.5, C = 85.5
Triangle 2: a = 36, b = 27.5, c = 44, A = 31, B = 116.5, C = 33.5
What are the possible triangles that satisfy the given conditions?Generally, To use the Law of Sines to solve for the possible triangles that satisfy the given conditions, we need to know at least one additional angle or side length of the triangle. The Law of Sines states that:
a/sin(A) = b/sin(B) = c/sin(C)
where a, b, and c are the side lengths of the triangle and A, B, and C are the corresponding angles opposite those sides. Therefore, we need to know at least one additional angle or side length in order to use the Law of Sines to solve for the remaining sides and angles of the triangle.
If we let B1 and B2 be the two possible values of angle B, then we can use the Law of Sines to find the corresponding values of angles A1 and A2 and sides a1 and a2 as follows:
a1/sin(A) = c/sin(B1) a2/sin(A) = c/sin(B2)
Solving these equations for a1 and a2, we get:
a1 = c * sin(A) / sin(B1) a2 = c * sin(A) / sin(B2)
Since we are given that a = 36 and c = 44, we can substitute these values into the above equations to find the values of a1 and a2.
a1 = 44 * sin(31) / sin(B1) a2 = 44 * sin(31) / sin(B2)
Now, we can use the Law of Sines to find the corresponding values of angles A1 and A2.
A1 = arcsin(a1 * sin(B1) / c) A2 = arcsin(a2 * sin(B2) / c)
Substituting the values of a1 and a2 that we found earlier and the given value of c, we get:
A1 = arcsin(36 * sin(B1) / 44) A2 = arcsin(36 * sin(B2) / 44)
To find the possible values of angles B1 and B2, we can use the fact that the sum of the angles in a triangle is always 180 degrees. Therefore, we have:
B1 + A1 + C1 = 180 degrees B2 + A2 + C2 = 180 degrees
Substituting the values of A1 and A2 that we found earlier, we get:
B1 + arcsin(36 * sin(B1) / 44) + (180 - 31 - B1) = 180 degrees B2 + arcsin(36 * sin(B2) / 44) + (180 - 31 - B2) = 180 degrees
Solving these equations for B1 and B2, we find that the possible values of B1 and B2 are:
B1 = 63.5 degrees B2 = 116.5 degrees
Therefore, the possible triangles that satisfy the given conditions are:
Triangle 1: a = 36, b = 43.5, c = 44, A = 31, B = 63.5, C = 85.5
Triangle 2: a = 36, b = 27.5, c = 44, A = 31, B = 116.5, C = 33.5
Note that these values should be rounded to one decimal place as requested.
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If the slant height of a triangular face of a square based pyramid of side
16cm is 17cm, find the volume of the pyramid.
Answer must be 1280 cube cm
Answer:
1280 cm³
I presume you are looking for the solution steps and these are below
Step-by-step explanation:
If the each side of the square pyramid is 16 and the slant height is 17, then the height of the pyramid, half the side and the slant edge form a right triangle.
The height of the pyramid can be computed using the Pythagorean theorem
Let a be each side, h the height and s the slant height
[tex]s^2 = h^2 + (a/2)^2\\\\\\s^2 = h^2 + \dfrac{a^2}{4}\\\\\textrm{Plugging in known values,}\\17^2 = h^2 + \dfrac{16^2}{4}\\\\289= h^2 + 64\\\\h^2 = 289 - 64\\\\h^2 = 225\\\\h = \sqrt{225} = 15\\\\[/tex]
The volume, V of a square pyramid of side a and height h is given by the formula:
[tex]V = \dfrac{1}{3}a^2h\\\\\\\textrm{Plugging in values,}\\\\V = \dfrac{1}{3}\cdot 16^2 \cdot 15\\\\[/tex]
[tex]V = \dfrac{1}{3} \cdot 256 \cdot 15\\\\V = 256 \cdot 5\\\\V = 1280[/tex]
The volume of the pyramid is 1280 cubic cm.
Given the slant height of the pyramid is 17 cm and the length of square base of the pyramid is 16 cm.
Now we know the volume of the pyramid with a square base = [tex]\frac{1}{3}[/tex]×area of the base×height.
So we need the values for area of the base and height of the pyramid.
Area of the square base=16×16 sq. cm.= 256 sq. cm.
Again we can apply the relation to find the value of height,
(Slant height)² = (Height)² + ([tex]\frac{1}{2}[/tex]×Length of the base)²
So, putting the values we get,
[tex](17)^2 = (height)^2 +(\frac{16}{2})^2[/tex]
⇒ [tex]289 = (height)^2+64[/tex]
⇒ [tex](height)^2= 225[/tex]
⇒ [tex]height = 15cm[/tex]
Now by appyling the volume of the pyramid we get,
Volume = [tex]\frac{1}{3} \times 256 \times 15[/tex] cubic cm. = 1280 cubic cm.
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The numbers below to put them in order from least to greatest: -19 13 -15 -8 -11 -13
The arrangement of the given numbers in ascending order is -19, -15, -13, -11, -8, 13
What is Ascending Order?
Ascending order refers to the arrangement of numbers in ascending order, from least to greatest. We must first compare numbers before we may arrange them in any order. Compare first, then order. Putting the numerals in ascending order: Count how many digits are in each number.
Solution:
Given the order is -19, 13, -15, -8, -11, -13
Arranging them in the Ascending Order
-19, -15, -13, -11, -8, 13
In the above arrangement the given numbers are arranged into ascending order
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What is the monthly payment for a $11,000 loan for 6 years with an annual interest rate of 3.1%
The monthly payment for the $11,000 loan for 6 years with an annual interest rate of 3.1% is $181.20.
How to calculate the monthly payment?From the information, we have a $11,000 loan for 6 years with an annual interest rate of 3.1%. The interest will be:
= Principal × Rate × Time.
= $11000 × 3.1% × 6.
= $2046
The monthly payment will be:
= ($11000 + $2046) / (6 × 12)
= $13046 / 72
= $181.20
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Write an equation of the line that passes through (-1,3) and is parallel to the line y=-3x+2
Slope-intercept form: [tex]y=mx+b[/tex]
m = slopeb = y-interceptParallel lines always have the same slopes and different y-intercepts.
Solving the QuestionWe're given:
Parallel to [tex]y=-3x+2[/tex]Passes through (-1,3)Because parallel lines have the same slopes, this line therefore has a slope of -3.
[tex]y=-3x+b[/tex]
Now, plug in the given point to solve for b:
[tex]3=-3(-1)+b\\3=3+b\\b=0[/tex]
Therefore:
[tex]y=-3x[/tex]
Answer[tex]y=-3x[/tex]
Find the Limit of the following given Function
The answer of the limit is -3
Paris and Jen are sisters that are away at different colleges. After a 5-hour drive, Paris is 324 miles from home, and after 7 hours she's 200 miles from home. Jen's trip home can be represented by the equation y = -58+422. Which statements are true? Select all that apply Jen is driving slower than Paris. Parts is driving slower than Jen Paris's school is farther from home than Jer's school Jen's school is farther from home than Paris's school It will take Paris longer to get home, even though she's driving faster
The correct statement is Jen is that driving faster than Paris.
Jen's journey home is represented by the equation y=58x+422.
Paris covers ,
324 miles in 5 hour
Therefore, her speed is,
324/5
= 64.8 miles/hr
Jen's rate is expressed as, y=58x+422.
Here , distance covered is 422 miles.
After comparing their speed we can conclude that , Jen is faster.
Distance word problems are a sort of algebra word problem that is very prevalent. They involve a scenario in which you must determine how quickly, far, or long one or more things have gone.
These are frequently referred to as train problems since one of the most well-known varieties of distance problems involves determining when two trains travelling in opposite directions cross paths.
Distance, pace, and time are the three fundamental components of movement and travel.
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Suppose that the consumer expenditure E (in dollars) depends on the market price p per unit (in dollars) according toE = 6,400p − 80p2.a)Because E represents consumer expenditure, only values of E ≥ 0 have meaning. For what p-values is E ≥ 0? (Enter your answer using interval notation.)
As per the given expenditure, the p value is 80
The term p value refers a number, calculated from a statistical test, that describes how likely you are to have found a particular set of observations if the null hypothesis were true.
Here we have given that suppose that the consumer expenditure E (in dollars) depends on the market price p per unit (in dollars) according to
E = 6,400p − 80p2.
where E represents consumer expenditure, only values of E ≥ 0 have meaning.
And we need to find value of P.
Through the given question we have identified the function of the following,
=> E = 6,400p − 80p²
Here we have to take p as common variable then we get,
=> E = p (6400 - 80p)
Now, take 80 as the common value then we get
=> E = 80p(80 - p)
When we equate this one with zero, then ew get
=> 80p(80-p) = 0
=> 80 - p = 0
Therefore, p is 80.
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16. STEM Connection Maya has 3.8 liters of a
solution. She needs 4.3 times more than she
already has. About how many liters of the solution
does she need?
Maya has a solution that volume is 3.8 liters. She needs more 20.12 liters solution.
What is solution?
A particular kind of homogenous mixture made up of two or more substances is known as a solution. A solute is a substance that has been dissolved in a solvent, which is the other substance in the mixture.
Given that Maya has a solution with volume 3.8 liters.
The symbol is used in mathematics to represent the multiplication operation and the resultant product. It is sometimes referred to as the times sign or the dimension sign.
Times means multiplication.
She requires 4.3 times more than she currently possesses..
She needs = 4.3 × 3.8 = 16.32 liters.
The total amount she needs is 16.32 + 3.8
Addition process:
1
16.32
+3.80
______
20.12
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