Question: The heights of adult men can be approximated as normal, with a mean of 70 and a standard deviation of 3, the probability that a man is shorter than 65 inches is approximately 0.0475 or 4.75%.
Let X be the height of an adult man, which follows a normal distribution with mean μ = 70 and standard deviation σ = 3. Then, we need to find the probability that a man is shorter than some height, say x₀. We can write this probability as P(X < x₀).To find P(X < x₀), we need to standardize the random variable X by subtracting the mean and dividing by the standard deviation. This yields a new random variable Z with a standard normal distribution. Mathematically, we can write this transformation as:Z = (X - μ) / σwhere Z is the standard normal variable.
Now, we can find P(X < x₀) as:P(X < x₀) = P((X - μ) / σ < (x₀ - μ) / σ) = P(Z < (x₀ - μ) / σ)Here, we use the fact that the probability of a standard normal variable being less than some value z is denoted as P(Z < z), which is available in standard normal tables.
Therefore, to find the probability that a man is shorter than some height x₀, we need to standardize the height x₀ using the mean μ = 70 and the standard deviation σ = 3, and then look up the corresponding probability from the standard normal table.In other words, the probability that a man is shorter than x₀ can be expressed as:P(X < x₀) = P(Z < (x₀ - 70) / 3)We can now use standard normal tables or software to find the probability P(Z < z) for any value z.
For example, if x₀ = 65 (i.e., we want to find the probability that a man is shorter than 65 inches), then we have:z = (65 - 70) / 3 = -1.67Using a standard normal table, we can find that P(Z < -1.67) = 0.0475. Therefore, the probability that a man is shorter than 65 inches is approximately 0.0475 or 4.75%. Thus, P(X < 65) = 0.0475 or 4.75%. Therefore, the probability that a man is shorter than 65 inches is approximately 0.0475 or 4.75%.
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b) There are x number of books that worth Rs. 35 each and 5 books worth Rs. 30 each in a parcel prepared as a gift. The value of two such parcels is Rs. 580. i. Build up an equation using the above information. ii. Find the value of x by solving the equation.
Answer:
Equation: 2(357+30×5) = 580
x=4
Step-by-step explanation:
In one package, there is such a relationship:
357+30X5 = y
(Y is the price of a package)
The price of two parcels is 580:
then. 24=580
y= 290
x=4, so: equation: 2(35x+150) =580
Step-by-step explanation:
A shopkeeper buys a number of books for Rs. 80. If he had bought 4 more for the same amount each book would have cost Rs. 1 less. How many books did he buy?
A
8
B
16
Correct Answer
C
24
D
28
Medium
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Updated on : 2022-09-05
Solution

Verified by Toppr
Correct option is B)
Let the shopkeeper buy x number of books.
According to the given condition cost of x books =Rs80
Therefore cost of each book =x80
Again when he had brought 4 more books
Then total books in this case =x+4
So cost of each book in this case =x+480
According to Question,
x80−x+480=1
x(x+4)80(x+4)−80x=1
x2+20x−16x−320=0
(x−16)(x+20)=0
x=16orx=−20
Hence the shopkeeper brought 16 books
Question 4(Multiple Choice Worth 2 points)
(Irrational Numbers MC)
Order √50,-7.1.3-7 from least to greatest.
0 -7.1.-7. √50,23
O
0-71.-7.7.23,√50
O
0 -7.1.-723√50
0-7-7.1,√50,23
Answer:
D
Step-by-step explanation:
The square root of 50 is approximately equal to 7.07
-7.1111… can be rounded to -7.11
23/3 is equal to approximately 7.67
-7 1/5 is equal to -7.2
Pablo needs to memorize words on a vocabulary list for Latin class he has 12 words to memorize and he is 3/4 done how many words has Pablo memorized so far
Answer:
9 words
Step-by-step explanation:
We know
He has 12 words to memorize, and he is 3/4 done.
How many words has Pablo memorized so far?
We Take
12 x 3/4 = 9 words
So, Pable has memorized 9 words.
1(1/2)= 1 1/2 draw number line and represent this
|-----|-----|-----|----|-----|-----|--│--|-----|----|-----|
-5 -4 -3 -2 -1 0 1 │ 2 3 4 5
1 1/2
On this number line, the tick mark labeled "1 1/2" is located halfway between the integer values of 1 and 2.
To represent the number 1 1/2 on a number line, we need to draw a horizontal line with evenly spaced tick marks. Each tick mark represents a specific value on the number line. Since 1 1/2 is a mixed number that includes a whole number (1) and a fraction (1/2), we need to locate it between the integer values of 1 and 2. The tick mark for 1 1/2 should be halfway between these two integers, which means it would be located at the midpoint of the line segment that connects the tick marks for 1 and 2. By placing the tick mark for 1 1/2 in the correct position on the number line, we can accurately represent this number visually.
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Find the value of x.
Answer:
x=1.9
Step-by-step explanation:
[tex]\frac{x}{4.6} =\frac{4.6}{11}[/tex]
[tex]11x=21.16[/tex]
[tex]X=1.9[/tex]
At which values in the interval [0, 2π) will the functions f (x) = 2cos2θ and g(x) = −1 − 4cos θ − 2cos2θ intersect?
a: theta equals pi over 3 comma 4 times pi over 3
b: theta equals pi over 3 comma 5 times pi over 3
c: theta equals 2 times pi over 3 comma 4 times pi over 3
d: theta equals 2 times pi over 3 comma 5 times pi over 3
The values in the interval [0, 2π) for which the two points would intersect as required is; Choice C; theta equals 2 times pi over 3 comma 4 times pi over 3.
What values of θ make the two functions intersect?Recall from the task content; the given functions are;
f (x) = 2cos2θ and g(x) = −1 − 4cos θ − 2cos2θ
Therefore, for intersection; f (θ) and g(θ):
2 cos²θ = −1 − 4cos θ − 2cos²θ
4cos²θ + 4cosθ + 1 = 0
let cos θ = y;
4y² + 4y + 1 = 0
y = -1/2
Therefore; -1/2 = cos θ
θ = cos-¹ (-1/2)
θ = 2π/3, 4π/3.
Ultimately, the correct answer choice is; Choice C; theta equals 2 times pi over 3 comma 4 times pi over 3.
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Write 735 as the product of its prime factor.
Answer:
[tex]735 = 3 \times 5 \times {7}^{2} [/tex]
Step-by-step explanation:
[tex]735 = 7 \times 105[/tex]
[tex]735 = 7 \times 3 \times 35[/tex]
[tex]735 = 7 \times 3 \times 5 \times 7[/tex]
[tex]735 = 3 \times 5 \times {7}^{2} [/tex]
the product of 2 rational numbers is 16/3.If one of the rational number is -26/3,find the other rational number
Answer:
- [tex]\frac{8}{13}[/tex]
Step-by-step explanation:
let n be the other rational number , then
- [tex]\frac{26}{3}[/tex] n = [tex]\frac{16}{3}[/tex]
[a number × its reciprocal = 1 ]
multiply both sides by the reciprocal - [tex]\frac{3}{26}[/tex]
n = [tex]\frac{16}{3}[/tex] × - [tex]\frac{3}{26}[/tex] ( cancel the 3 on numerator/ denominator )
n = - [tex]\frac{16}{26}[/tex] = - [tex]\frac{8}{13}[/tex]
Find the interest refund on a 35-month loan with interest of $2,802 if the loan is paid in full with 13 months remaining.
Answer: $1,071.54
Step-by-step explanation:
To find the interest refund, first we need to calculate the total interest charged on the loan. We can do this by multiplying the monthly interest by the number of months in the loan:
Monthly interest = Total interest / Number of months
Monthly interest = $2,802 / 35
Monthly interest = $80.06
Total interest charged on the loan = Monthly interest x Number of months
Total interest charged on the loan = $80.06 x 35
Total interest charged on the loan = $2,802.10
Now we need to calculate the interest that would have been charged for the remaining 13 months of the loan:
Interest for remaining 13 months = Monthly interest x Remaining months
Interest for remaining 13 months = $80.06 x 13
Interest for remaining 13 months = $1,040.78
Finally, we can find the interest refund by subtracting the interest for the remaining 13 months from the total interest charged on the loan:
Interest refund = Total interest charged - Interest for remaining months
Interest refund = $2,802.10 - $1,040.78
Interest refund = $1,074.32
Therefore, the interest refund on the loan is $1,074.30.
If 140 men working 10 hours a day can build a house in 16 days, find out how many men will build same kind of house in 12 days by working 13 hours a day?
We need 144 men to build the house in 12 days working 13 hours a day.
Let M be the number of men needed to build the house in 12 days working 13 hours a day.
140 x 10 x 16 = M x 13 x 12
Simplifying the equation, we get:
22400 = 156M
Dividing both sides by 156, we get:
M = 144.1
An equation in mathematics is a statement that two expressions are equal. It consists of two sides, the left-hand side (LHS) and the right-hand side (RHS), separated by an equal sign (=). The expressions on either side can be numbers, variables, or combinations of both. The equation expresses that the values of the expressions on both sides are equivalent.
Equations play a fundamental role in many areas of mathematics and are used to model various real-world situations, such as physics, engineering, and finance. They can be solved using various techniques, such as substitution, elimination, or graphing, to find the values of the variables that satisfy the equation.
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A triangle has two sides of length 3 and 16. What is the largest possible whole-number length for the third side
The largest possible whole-number length for the third side is 18, which satisfies all three inequalities.
What is inequality theorem?The triangle inequality theorem explains the relationship between the three sides of a triangle. This theorem states that for any triangle, the sum of the lengths of the first two sides is always larger than the length of the third side.
According to question:Let x be the length of the third side. By the triangle inequality, we have:
3 + 16 > x and 16 + x > 3 and 3 + x > 16
Simplifying, we get:
19 > x and x > 13 and x < 19
The largest possible whole-number length for the third side is 18, which satisfies all three inequalities.
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Reduce each expression to a polynomial
((y-b)^(2))/(y-b+1)+(y-b)/(y-b+1)
The given expression ((y-b)²/(y-b+1)+(y-b)/(y-b+1) after being reduced to a polynomial, can be represented as y-b.
In order to reduce the given equation to a polynomial, we are required to simplify and combine like terms. First, we can simplify the expression in the numerator by expanding the square:
((y-b)²/(y-b+1) = (y-b)(y-b)/(y-b+1) = (y-b)²/(y-b+1)
Now, we can combine the two terms in the equation by finding a common denominator:
(y-b)²/(y-b+1) + (y-b)/(y-b+1) = [(y-b)² + (y-b)]/(y-b+1)
Next, we can combine the terms in the numerator by factoring out (y-b):
[(y-b)² + (y-b)]/(y-b+1) = (y-b)(y-b+1)/(y-b+1)
Finally, we can cancel out the common factor of (y-b+1) in the numerator and denominator to get the polynomial:
(y-b)(y-b+1)/(y-b+1) = y-b
Therefore, the equation ((y-b)²)/(y-b+1)+(y-b)/(y-b+1) after being simplified, is equivalent to the polynomial y-b.
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Suppose that s is the position function of an object, given as s(t) = 2t - 7. We compute the instantaneous velocity of the object at t = 6 as follows. Use exact values. First we compute and simplify (6 +h). s(6 + h) = Then we compute and simplify the average velocity of the object between t = 6 and t = 6 + h. 8(6+h) - s(6) h = Rationalize the numerator in the average velocity. (If it applies, simplify again.) $(6 + h) - $(6) h The instantaneous velocity of the object att = 6 is the limit of the average velocity as h approaches zero. s(6 + h) – $(6) v(6) lim h -0
The instantaneous velocity of the object at t = 6 is 2.
Suppose that s is the position function of an object, given as s(t) = 2t - 7. We compute the instantaneous velocity of the object at t = 6 as follows. Use exact values. First we compute and simplify (6 + h). s(6 + h) = 2(6 + h) - 7 = 12 + 2h - 7 = 2h + 5Then we compute and simplify the average velocity of the object between t = 6 and t = 6 + h.8(6+h) - s(6) h = 8(6 + h) - (2(6) - 7) h= 8h + 56
Then, to rationalize the numerator in the average velocity. (If it applies, simplify again.)$(6 + h) - $(6) h(h(h) + 56)/(h(h)) = (8h + 56)/h The instantaneous velocity of the object att = 6 is the limit of the average velocity as h approaches zero.s(6 + h) – $(6) v(6) lim h -0s(6 + h) – s(6) v(6) lim h -0Using the above calculation, we get:s(6 + h) – s(6) / h lim h -0s(6 + h) = 2(6 + h) - 7 = 2h + 5So,s(6 + h) – s(6) / h lim h -0(2h + 5 - (2(6) - 7)) / h= (2h + 5 - 5) / h = (2h / h) = 2
Therefore, the instantaneous velocity of the object at t = 6 is 2.
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Brittany needed new tires for her truck. She went to the auto shop and bought 4 tires on sale for $85.95 each. The salesman told her that she saved a total of $96.16. If Brittany saved the same amount on each tire, what was the original price of each tire?
The best solution gets brainlist
Answer:
$109.99
Step-by-step explanation:
The original price of each tire is [tex]\[/tex][tex]109.99[/tex]
Solution:Take the amount saved and divide by 4 to find the amount saved on each tire
[tex]96.16\div4 =24.04[/tex]
Add that to the sale price of each tire to find the original price
[tex]85.95+24.04 =109.99[/tex]
Therefore, The original price is $109.99.
Write the first four terms of the sequence defined by a n = 5
{5, if n=1
a n -1 -5, if n>1
Answer:
The sequence is defined as follows:
a1 = 5
an = an-1 - 5, for n > 1
Using this definition, we can find the first four terms of the sequence as follows:
a1 = 5
a2 = a1 - 5 = 5 - 5 = 0
a3 = a2 - 5 = 0 - 5 = -5
a4 = a3 - 5 = -5 - 5 = -10
Therefore, the first four terms of the sequence are: 5, 0, -5, -10.
22 The regular selling price is a 22" tube television is $389. The markdown rate is 33%. Use the
percent paid to determine the sale price.
A. $245.34
C. $260.63
B. $267.89
D. $287.56
The Sale price is C. $260.63.
What is selling price?Selling price is the price at which a product or service is sold by a business or seller to a customer. It is the amount of money that a customer must pay in order to purchase the product or service. The selling price is typically determined by factors such as production costs, competition, supply and demand, and profit margins.
What is sale price?Sale price is the discounted price at which a product or service is sold for a limited period of time. It is usually a lower price than the regular price, and it is offered to customers as an incentive to make a purchase. Sale prices can be determined by applying a discount or markdown to the regular selling price.
In the given question,
To find the sale price, we need to first calculate the amount of the markdown:
Markdown = Regular Price x Markdown Rate
Markdown = $389 x 0.33
Markdown = $128.37
The sale price is then the regular price minus the markdown:
Sale Price = Regular Price - Markdown
Sale Price = $389 - $128.37
Sale Price = $260.63
Therefore, the answer is C. $260.63.
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please help me with this savvas question!
Therefore, the compound inequality for the diameter of the washers is: 3.150 ≤ d ≤ 3.240.
What is inequality?In mathematics, an inequality is a statement that compares two values or expressions, indicating that one is greater than, less than, or equal to the other. The symbols used to represent inequalities are:
">" which means "greater than"
"<" which means "less than"
"≥" which means "greater than or equal to"
"≤" which means "less than or equal to"
Inequalities can be solved by applying algebraic techniques, such as adding, subtracting, multiplying, or dividing both sides of the inequality by the same number. The solution to an inequality is a range of values that satisfy the inequality.
Here,
The formula for the circumference of a circle in terms of its diameter is:
C = πd
where π (pi) is approximately 3.14.
We are given that the acceptable range for the circumference of the washer is 9.9 ≤ C ≤ 10.2 centimeters. Substituting C = 3.14d into this inequality, we get:
9.9 ≤ 3.14d ≤ 10.2
Dividing all sides of the inequality by 3.14, we obtain:
3.15 ≤ d ≤ 3.24
Rounding to three decimal places, the corresponding interval for the diameters of the washers is:
3.150 ≤ d ≤ 3.240
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If a certain apple tree grew 2 feet and then tripled its height, it would become 4 feet
shorter than the pine tree that grows on the other end of the street. Which
of the formulas below describes the relation between the height of the apple tree a
and the height of the pine tree p?
A) P-4=3a+2
B) P=2(a+3)+4
C) P=3(a+2)-4
D) P=3a+10
Answer:
Step-by-step explanation:
C.) P = 3(a+2)-4
The formula which describes the relation between the height of the apple tree and the height of the pine tree p is P=3(a+2)-4, the correct option is C.
What is a linear equation?A linear equation is an equation that has the variable of the highest power of 1. The standard form of a linear equation is of the form Ax + B = 0.
We are given that;
Growth of apple tree= 2feet
Now,
Let's call the original height of the apple tree "h". According to the problem, if the apple tree grew 2 feet and then tripled its height, it would become 4 feet shorter than the pine tree. So we can write:
3(h+2) - 4 = p
Simplifying, we get:
3h + 2 = p
Now we can see that option (D) P=3a+10 is very similar to our expression, but it has a constant term of 10 instead of 2. This constant term does not match the problem statement, which says that the apple tree would be 4 feet shorter than the pine tree, not taller. Therefore, option (D) is not the correct answer.
Option (A) P-4=3a+2 also does not match the problem statement. If we solve for p, we get:
P = 3a + 6
This means that the apple tree would be 6 feet shorter than the pine tree, not 4 feet shorter as stated in the problem.
Option (B) P=2(a+3)+4 also does not match the problem statement. If we solve for p, we get:
P = 2a + 10
This means that the apple tree would be 10 feet shorter than the pine tree, not 4 feet shorter as stated in the problem.
Option (C) P=3(a+2)-4 matches our expression from earlier. If we solve for p, we get:
P = 3a + 2
Therefore, by equation the answer will be P=3(a+2)-4.
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Write the equation of a line that is perpendicular to y=½x - 9 and passes through the point (3, -2).
Answer:
y = - 2x + 4
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
y = [tex]\frac{1}{2}[/tex] x - 9 ← is in slope- intercept form
with slope m = [tex]\frac{1}{2}[/tex]
given a line with slope m then the slope of a line perpendicular to it is
[tex]m_{perpendicular}[/tex] = - [tex]\frac{1}{m}[/tex] = - [tex]\frac{1}{\frac{1}{2} }[/tex] = - 2 , then
y = - 2x + c ← is the partial equation
to find c substitute (3, - 2 ) into the partial equation
- 2 = - 2(3) + c = - 6 + c ( add 6 to both sides )
4 = c
y = - 2x + 4 ← equation of perpendicular line
Suppose Z follows the standard normal distribution. Calculate the following probabilities using the ALEKS calculator. Round your responses to at least three decimal places. (a) P(Z < 0.79) = Х 5 ? (b) P(Z > 0.75) (c) P(-1.06 < Z< 2.17) =
The probabilities Z > 0.75 is P(Z > 0.75) = 1 - P(Z < 0.75).
The probability of Z > 0.75 is 1 - 0.77337 = 0.22663
The probability of Z < -1.06 from it. P(-1.06 < Z< 2.17) = P(Z < 2.17) - P(Z < -1.06) = 0.98425 - 0.14457 = 0.83968
Suppose Z follows the standard normal distribution. The probabilities using the ALEKS calculator are given below.(a) P(Z < 0.79) = 0.78524. (rounded to 5 decimal places)(b) P(Z > 0.75) = 1 - P(Z < 0.75) = 1 - 0.77337 = 0.22663. (rounded to 5 decimal places)(c) P(-1.06 < Z< 2.17) = P(Z < 2.17) - P(Z < -1.06) = 0.98425 - 0.14457 = 0.83968. (rounded to 5 decimal places). In the standard normal distribution, the mean is equal to zero and the standard deviation is equal to 1. The notation for a standard normal random variable is z. Z is a random variable with a standard normal distribution and P(Z) denotes the probability of the random variable Z. Suppose z follows a standard normal distribution then the probability of Z < 0.79 is P(Z < 0.79) = 0.78524. So, the answer is 0.78524(rounded to 5 decimal places).Suppose z follows a standard normal distribution then the probability of Z > 0.75 is P(Z > 0.75) = 1 - P(Z < 0.75). Therefore, the probability of Z > 0.75 is 1 - 0.77337 = 0.22663(rounded to 5 decimal places).Therefore, the probability of -1.06 < Z< 2.17 can be found by finding the probability of Z < 2.17 and then subtracting the probability of Z < -1.06 from it. P(-1.06 < Z< 2.17) = P(Z < 2.17) - P(Z < -1.06) = 0.98425 - 0.14457 = 0.83968(rounded to 5 decimal places).
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Can i get assistance with this?
Answer:
see attached
Step-by-step explanation:
You want the given triangle dilated by a factor of -3 about point A.
DilationTo find the image point corresponding to a pre-image point, multiply the pre-image point's distance from A by the dilation factor. The negative sign means the distance to the image point is measured in the opposite direction.
In the attached figure, the chosen point is 4 units up and 5 units right of A. Its image in the dilated figure is 3·4 = 12 units down, and 3·5 = 15 units left of A.
This same process can be used to locate the other vertices of the triangle's image.
All the students in the sixth grade either purchased their lunch or brought their lunch from home on Monday.
• 24% of the students purchased their lunch.
• 190 students brought their lunch from home.
How many students are in the sixth grade?
The number of students that are in the sixth grade is given as follows:
250 students.
How to obtain the number of students?The number of students is obtained applying the proportions in the context of the problem.
We know that all students in the sixth grade either purchased their lunch or brought their lunch from home on Monday, and 24% of the students purchased their lunch, hence 76% of the students brought their lunch from home.
190 students brought their lunch from home, which is equivalent to 76% of the number of students, hence the number of students is given as follows:
0.76n = 190
n = 190/0.76
n = 250 students.
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Does someone mind helping me with this problem? Thank you!
the answer to the problem that you need to is 1024
Shade in the regions represented by the inequalities
Answer:
Step-by-step explanation:
see diagram
Xavier buys a dog collar that costs $6.79. He pays for the dog collar
with a $10 bill.
How much change does Xavier receive?
Answer: Xavier will receive $3.21 in change.
Step-by-step explanation:
To find the change Xavier receives, we need to subtract the cost of the dog collar from the amount he paid with his $10 bill:
Change = $10 - $6.79 = $3.21
Therefore, Xavier will receive $3.21 in change.
some rate functions require algebraic manipulation or simplification to set the stage for undoing the chain rule or other antiderivative techniques. find an equivalent closed form for each function.a. S π / π /4 5t+4 / t² + 1 dtHint : begin by writing as a sum of two functions ____ previewb. S π/t 4tan (t) dt Hint : begin by using a trig identity to change the form of the rate function___ preview
the given form of the rate function:[tex]tan² (t) + 1 = sec²[/tex](t)
Therefore, we can write the given function as:c (t) dtUsing integration by substitution, we haveu = tan (t) ⇒ du = sec² (t) dt
Therefore,S [tex]π/t tan (t) sec² (t) dt= S u du= ln |tan (t)| + C[/tex]Thus, the equivalent closed form of the given function is:S π/t 4tan (t) dt= 4 ln |tan (t)| + C
a. S π/π/4 5t+4/t² + 1 dt equivalent closed formThe question demands to find an equivalent closed form for each function. So let's find the equivalent closed form for the given functions:a. S π/π/4 5t+4/t² + 1 dt
Hint: begin by writing as a sum of two functionsNow, we need to write the given function as a sum of two functions. Let's first write the numerator of the function as a sum of two functions.
Using the formula, a²-b² = (a+b)(a-b), we have5t + 4 = (2 + √21)(√21 - 2)Therefore, we can write the numerator of the function as follows:5t + 4 = (√21 - 2)² - 17Using this in the given function,
we haveπ/π/4 [(√21 - 2)² - 17]/t² + 1 dtLet's further simplify the numeratorπ/π/4 [21 + 4 - 4√21 - 17t² + 34t - 17] / (t² + 1) dt= π/π/4 [-17t² + 34t + 8 - 4√21]/(t² + 1) dtLet's now find the closed form of this function using the integration formulaS f(x) dx = ln |f(x)| + C Therefore, the equivalent closed form of the function is:
S π/π/4 5t+4/t² + 1 dt= π/π/4 [-17t² + 34t + 8 - 4√21]/(t² + 1) dt= - π/2 ln |t² + 1| + 34 π/2 arctan (t) - 17 π/2 t + 2 π/√21 arctan [(2t-√21)/ √21] + Cb. S π/t 4tan (t) dt equivalent closed formNow, let's find the equivalent closed form of the second given function.b. S π/t 4tan (t)
dtHint: begin by using a trig identity to change the form of the rate function Let's now use the following trig identity to change
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The annual salaries (in $) within a certain profession are modelled by a random variable with the cumulative distribution function F(x)= {1−kx^−3 for x>44000 {0 otherwise, for some constant k. For these problems, please ensure your answers are accurate to within 3 decimals. a)Find the constant k here and provide its natural logarithm to three decimal places. b)Calculate the mean salary given by the model.
a) The constant k is 5.427 x 10^−12 and its natural logarithm is -26.68.
b) The mean salary of the given model by using the probability density function is approximately $270.86.
a) The cumulative distribution function of the given random variable is provided as follows:
F(x) = {1−kx^−3 if x>44000, and 0 otherwise
The cumulative distribution function is given as
F(x) = 1−kx^−3 if x>44000 and F(x) = 0, if x≤44000i)
We need to check the value of the cumulative distribution function at 44000
We have, F(44000) = 0
0 = 1−k(44000)^−3
⇒ 1 = k(44000)^−3
⇒ k = 1/(44000)^−3
⇒ 5.427 x 10^−12
Taking the natural logarithm of k, we have ln(k) = −28.68 (approx.)
Hence, the constant k is 5.427 x 10^−12 and its natural logarithm to three decimal places is -28.68
b) The probability density function is given as,
f(x) = F'(x) = 3kx^−4, for x>44000 and f(x) = 0, otherwise
The mean or expected value of the random variable is given as
E(X) = ∫[−∞,∞]xf(x)dx
= ∫[44000,∞]x(3kx^−4)dx
= 3k∫[44000,∞]x^−3dx
= 3k[(−1/2)x^−2] [∞,44000]
= (3k/2)(44000)^−2
= 270.86 (approx.)
Therefore, the mean salary given by the model is $270.86 (approx.)
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If n(Ax B) = 72 and n(A) = 24, find n(B).
Solving for Cartesian product n(B), we have n(B) = 72 / 24 = 3.
What is Cartesian product?The Cartesian product is a mathematical operation that takes two sets and produces a set of all possible ordered pairs of elements from both sets.
In other words, if A and B are two sets, their Cartesian product (written as A × B) is the set of all possible ordered pairs (a, b) where a is an element of A and b is an element of B.
For example, if A = {1, 2} and B = {3, 4}, then A × B = {(1, 3), (1, 4), (2, 3), (2, 4)}.
By the question.
We know that n (Ax B) represents the number of elements in the set obtained by taking the Cartesian product of sets A and B.
Using the formula for the size of the Cartesian product, we have:
n (Ax B) = n(A) x n(B)
Substituting the given values, we get: 72 = 24 x n(B)
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Which exspression is equivalent to 9(4/3m-5-2/3m+2)
By answering the presented question, we may conclude that Therefore, the expression 9(4/3m-5-2/3m+2) is equivalent to 6m - 27.
what is expression ?An expression in mathematics is a collection of representations, numbers, and conglomerates that mimic a statistical correlation or regularity. A real number, a mutable, or a mix of the two can be used as an expression. Mathematical operators include addition, subtraction, fast spread, division, and exponentiation. Expressions are often used in arithmetic, mathematic, and form. They are used in the representation of mathematical formulas, the solution of equations, and the simplification of mathematical relationships.
To simplify the expression,
[tex]a(b+c) = ab + ac\\9(4/3m-5-2/3m+2) = 9(4/3m - 2/3m - 5 + 2)\\= 9(2/3m - 3)\\= 6m - 27[/tex]
Therefore, the expression 9(4/3m-5-2/3m+2) is equivalent to 6m - 27.
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A surfboard is in the shape of a rectangle and semicircle. The perimeter is to be 4m. Find the maximum area of the surfboard correct to 2 places.
The maximum area of the surfboard correct to 2 places is 0.67 m².
Given that a surfboard is in the shape of a rectangle and a semicircle, and its perimeter is to be 4m. We need to find the maximum area of the surfboard, correct to 2 decimal places.
Let the radius of the semicircle be 'r' and the length and breadth of the rectangle be 'l' and 'b' respectively. Perimeter of the surfboard = [tex]4m => l + 2r + b + 2r = 4 => l + b = 4 - 4r[/tex] -----(1)
Area of surfboard = Area of rectangle + Area of semicircle Area of rectangle = l × b Area of semicircle = πr²/2 + 2r²/2 = (π + 2)r²/2Area of surfboard = l × b + (π + 2)r²/2 -----(2)
We have to maximize the area of the surfboard. So, we have to find the value of 'l', 'b', and 'r' such that the area of the surfboard is maximum .From equation (1), we have l + b = 4 - 4r => l = 4 - 4r - bWe will substitute this value of 'l' in equation (2)
Area of surfboard = l × b + (π + 2)r²/2 = (4 - 4r - b) × b + (π + 2)r²/2 = -2b² + (4 - 4r) b + (π + 2)r²/2Now, we have to maximize the area of the surfboard, that is, we need to find the maximum value of the above equation.
To find the maximum value of the equation, we can differentiate the above equation with respect to 'b' and equate it to zero. d(Area of surfboard)/db = -4b + 4 - 4r = 0 => b = 1 - r Substitute the value of 'b' in equation (1),
we get l = 3r - 3Now, we can substitute the values of 'l' and 'b' in the equation for the area of the surfboard.
Area of surfboard =
[tex]l × b + (π + 2)r²/2 = (3r - 3)(1 - r) + (π + 2)r²/2 = -r³ + (π/2 - 1)r² + 3r - 3[/tex]
[tex]-r³ + (π/2 - 1)r² + 3r - 3 = -0.6685 m² \\[/tex]
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