Answer:
Part A)
2048π/3 cubic units.
Part B)
8192π/15 units.
Step-by-step explanation:
We are given that R is the finite region bounded by the graphs of functions:
[tex]f(x)=4\sqrt{x}\text{ and } g(x)=x[/tex]
Part A)
We want to find the volume of the solid of revolution obtained when rotating R about the x-axis.
We can use the washer method, given by:
[tex]\displaystyle \pi\int_a^b[R(x)]^2-[r(x)]^2\, dx[/tex]
Where R is the outer radius and r is the inner radius.
Find the points of intersection of the two graphs:
[tex]\displaystyle \begin{aligned} 4\sqrt{x} & = x \\ 16x&= x^2 \\ x^2-16x&= 0 \\ x(x-16) & = 0 \\ x&=0 \text{ and } x=16\end{aligned}[/tex]
Hence, our limits of integration is from x = 0 to x = 16.
Since 4√x ≥ x for all values of x between [0, 16], the outer radius R is f(x) and the inner radius r is g(x). Substitute:
[tex]\displaystyle V=\pi\int_0^{16}(4\sqrt{x})^2-(x)^2\, dx[/tex]
Evaluate:
[tex]\displaystyle \begin{aligned} \displaystyle V&=\pi\int_0^{16}(4\sqrt{x})^2-(x)^2\, dx \\\\ &=\pi\int_0^{16} 16x-x^2\, dx \\\\ &=\pi\left(8x^2-\frac{1}{3}x^3\Big|_{0}^{16}\right)\\\\ &=\frac{2048\pi}{3}\text{ units}^3 \end{aligned}[/tex]
The volume is 2048π/3 cubic units.
Part B)
We want to find the volume of the solid of revolution obtained when rotating R about the y-axis.
First, rewrite each function in terms of y:
[tex]\displaystyle f(y) = \frac{y^2}{16}\text{ and } g(y) = y[/tex]
Solving for the intersection yields y = 0 and y = 16. So, our limits of integration are from y = 0 to y = 16.
The washer method for revolving about the y-axis is given by:
[tex]\displaystyle V=\pi\int_{a}^{b}[R(y)]^2-[r(y)]^2\, dy[/tex]
Since g(y) ≥ f(y) for all y in the interval [0, 16], our outer radius R is g(y) and our inner radius r is f(y). Substitute and evaluate:
[tex]\displaystyle \begin{aligned} \displaystyle V&=\pi\int_{a}^{b}[R(y)]^2-[r(y)]^2\, dy \\\\ &=\pi\int_{0}^{16} (y)^2- \left(\frac{y^2}{16}\right)^2\, dy\\\\ &=\pi\int_0^{16} y^2 - \frac{y^4}{256} \, dy \\\\ &=\pi\left(\frac{1}{3}y^3-\frac{1}{1280}y^5\Bigg|_{0}^{16}\right)\\\\ &=\frac{8192\pi}{15}\text{ units}^3\end{aligned}[/tex]
The volume is 8192π/15 cubic units.
Find the volume
h=9cm
8cm
8cm
Answer: (8x8x9)/3=192
-09
2 1 point
The amount of a radioactive substance y that remains after t years is given by the equation y = a (e)^kt, where a is the initial
amount present and k is the decay constant for the radioactive substance. If a = 100, y = 50, and k = -0.035, find t.
Answer:
19.80
Step-by-step explanation:
Given the equation :
y = a (e)^kt
If a = 100, y = 50, and k = -0.035, find t.
50 = 100(e)^(-0.035t)
50/100 = e^(-0.035t)
0.5 = e^-0.035t
Take the In
In(0.5) = - 0.035t
-0.693147 = - 0.035t
-0.693147 / - 0.035 = t
19.8042 = t
Hence, t = 19.80
In the diagram, point D is the center of the medium-sized circle that passes through C and E, and it is also the center of the largest circle that passes through A and G. Each of the diameters of the small circles with centers B and F equals the radius of the medium-sized circle with center D. The shaded area is what fraction of the largest circle?Single choice.
9514 1404 393
Answer:
5/8
Step-by-step explanation:
The area of the smaller circles is proportional to the square of the ratio of their diameters. The two smallest circles have diameters equal to 1/4 the diameter of the largest circle. Hence their areas are (1/4)^2 = 1/16 of that of the largest circle.
Similarly, the medium circle has a diameter half that of the largest circle, so its area is (1/2)^2 = 1/4 of the are of the largest circle.
The smaller circles subtract 2×1/16 +1/4 = 3/8 of the area of the largest circle. Then the shading is 1-3/8 = 5/8 of the area of the largest circle.
5. In 2015, Texas led the nation in the percentage of people who lacked health insurance (21.6% of the population). It is known that, nationally, 5% of patients account for 50% of the costs of healthcare. These are the “high cost” patients Assume* that: Being a high cost patient and being uninsured are independent characteristics Insured and uninsured people become “patients” at the same rate The uninsured and high cost patients in Texas are evenly distributed across the state, and that high cost patients are evenly distributed across insured and uninsured patient populations a. What is the probability that a patient in a Texas healthcare facility will be a high cost patient who is uninsured?
Answer: 0.108
Step-by-step explanation:
Since the probability of the uninsured is 21.6% of the population, then the probability of insured will be:
= 1 - 21.6%
= 78.4%
The probability of high cost patients is 5%. Therefore, the probability that a patient in a Texas healthcare facility will be a high cost patient who is uninsured will be:
= 5% × 21.6%
= 0.05 × 0.216
= 0.108
Points A, B, C, and D lie on a line in that order. If AD/AC = 2/1 and AD/AB = 3/1, what is the value of AC/BD?
9514 1404 393
Answer:
3/4
Step-by-step explanation:
It might be easier to start by expressing the ratios with AD as the denominator.
AD/AC = 2/1 ⇒ AC/AD = 1/2
AD/AB = 3/1 ⇒ AB/AD = 1/3
From the latter, we have ...
(AD -AB)/AD = 1 -1/3 = 2/3 = BD/AD
Then the desired ratio is ...
AC/BD = (AC/AD)/(BD/AD) = (1/2)/(2/3) = (3/6)/(4/6)
AC/BD = 3/4
On a coordinate plane, 2 triangles are shown. The first triangle has points A (negative 1, negative 2), B (negative 4, negative 2), C (negative 1, negative 4). The second triangle has points A prime (1, 2), B prime (4, 2), C prime (1, 4). What rule describes the rotation about the origin? (x, y) → How many degrees was the figure rotated about the origin?
9514 1404 393
Answer:
(x, y) ⇒ (-x, -y)180°Step-by-step explanation:
Each image point has its signs reversed from the pre-image point.
(x, y) ⇒ (-x, -y) . . . . describes the rotation
Rotation from the third quadrant (A) to the first quadrant (A') is a rotation of 180°.
Answer:
3rd and 2nd option
Step-by-step explanation:
Midsegments geometry acellus pls helppfpfpff
Answer:
BC = 28
Step-by-step explanation:
The midsegment DF is half the measure of the third side BC , then
BC = 2 × DF = 2 × 14 = 28
The value of y varies with x and z, and y=8, when x=4 and z=10. What is the value of y when x=5 and z=11
Un automóvil consume 4 galones de gasolina al recorrer 180 kilómetros y para recorrer 900 kilómetros necesita 20 galones ¿cuántos kilómetros recorre por galón? ¿Cuantos galones consumirá en 2700 kilómetros?
Answer:
45 km por galón
60 galones en 2700 Km
Step-by-step explanation:
180 / 4
45 km por galón
900 / 45
20 galones
2700 / 45
60 galones en 2700 Km
4b^2+300=0 this is a quadratic equation that I am trying to solve including any solutions with imaginary numbers I will include a picture
Answer:
b= 5i square root of 3
b = -5i square root of 3
Step-by-step explanation:
4b^2+300=0
4b^2 = -300
b^2 = -75
b = square root of -75
b = -75^1/2
^1/2 means square root
b = 25^1/2 * 3^1/2 * i
b= 5i square root of 3
b = -5i square root of 3
The cardinal number of {200, 201, 202, 203, ..., 1099}
Answer:
I have not been able to answer it sorry
24
4
3+
2+
2
1
-3
-
-1
1
1
2
3
4
-1+
-2 +
-3+
4
What is the slope of the line?
Answer:
1.5/2
Step-by-step explanation:
slope formula = y2-y1/ x2 - x1
point one (2,0)
point 2 (0, 1.5)
you dont really need to subtract anything because the intercepts, so the slope is 1.5/2
(slope or m = 1.5 - 0 / 2 - 0 )
x intercept = value of x when y is 0
y intercept = value of y when x is 0
the slope of line is
Answer:
there is no file attached
Step-by-step explanation:
class 7th chapter: Simple Equation
The solution of the equation p-1 =20 is -------- *
a) 19
b) 20
c) 21
Answer:
C
Step-by-step explanation:
p=20+1
Find the numerical value of each expression. (Round your answers to five decimal places.) (a) sinh(ln(5)) (b) sinh(5)
sinh(ln(4)) = (exp(ln(4)) - exp(-ln(4)))/2 = (4 - 1/4)/2 = 15/8 = 1.875
sinh(4) = (exp(4) - exp(-4))/2 ≈ 27.28992
Value of the expression in which each variable was swapped out with a number from its corresponding domain sinh (l5)
How do you determine an expression's numerical value?sinh (5)
=sinh(1.6094) =2.39990 rad
=sinh(1.6094) =2.3
By doing the following, you may determine the numerical value of an algebraic expression: Replace each variable with the specified number. Then, enter your score in your team's table.
Analyze expressions that are linear.Multi-variable expressions should be evaluated.Analyze expressions that are not linear.Value of the expression in which each variable was swapped out with a number from its corresponding domain. In the case of a number with only one digit, referring to the numerical value associated with a digit by its "value" is a convenient shorthand.
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Hannah ran 12 laps for 8 days. How many laps did she run in total if she take a break of 1 complete day and 1 half day.
Answer:
The correct answer would be - 9.75 laps (if runs 12 laps in 8 days) or 78 laps (if 12 laps each day for 8 days)
Step-by-step explanation:
Given:
a) Laps covered in 8 days = 12
interval = 1 and half day
total laps = ?
Solution:
To know the total laps with intervals we need to calculate the laps run each day :
= 12/8 laps per day
= 3/2 laps per day
Now multiply the daily run with days
= (3/2)*6.5 (due to 8 - 1.5 = 6,5 days)
= 9.75 laps
B) Given:
Laps covered in 8 days = 12*8 =96
interval = 1 and half day
total laps = ?
Solution:
To know the total laps with intervals we need to calculate the laps run each day :
= 96/8 laps per day
= 12laps per day
Now multiply the daily run with days
= 12*6.5 (due to 8 - 1.5 = 6,5 days)
= 78 laps
will give brainyest (m^2/3 n^-1/3)^6
Step-by-step explanation:
here is the answer to your question
HELP ASAP I WILL GIVE BRAINLIST
Find the length of an arc of a circle with a 8-cm radius associated with a central angle of 240 degrees. Give your answer in exact and approximate form to the nearest hundredth. Show and explain your work
Answer:
33.51 cm
Step-by-step explanation:
240/360 = 2/3 (Arc length is 2/3 of the total circumference)
C = 2[tex]\pi[/tex]r ( Calculate the total circumference)
C = 2(8)[tex]\pi[/tex]
C = 50.265
2/3(50.265) (Take 2/3 of the circumference. times 2 divide by 3)
33.51
Use a calculator and leave the answer to C and then multiply and divide. You get a more precise answer.
The exact arc length is [tex]\frac{32\pi}{3}[/tex] radians.
The arc length in approximate form is 33.49 radians.
What is the formula for arc length?[tex]s = r\times \theta[/tex]
where r is the radius of the circle and [tex]\theta[/tex] is the central angle in radians.
How to convert angle from degrees to radians?Radians = Degrees ×[tex]\frac{\pi}{180^{\circ}}[/tex]
For given question,
We have been given a circle with a 8-cm radius associated with a central angle of 240 degrees.
[tex]r=8~cm,~\theta=240^{\circ}[/tex]
First we convert angle in radians.
[tex]\theta=240^{\circ}\\\\\theta=240^{\circ} \times \frac{\pi}{180^{\circ}}\\\\ \theta=\frac{4\pi}{3}[/tex]
Using the formula of the arc length,
[tex]s=8\times \frac{4\pi}{3} \\\\s=\frac{32\pi}{3}[/tex]
The exact answer of the arc length is [tex]s=\frac{32\pi}{3}[/tex]
Substitute the value of [tex]\pi = 3.14[/tex]
So, the arc length would be,
[tex]\Rightarrow s=\frac{32\times \pi}{3}\\\\\Rightarrow s=\frac{32\times 3.14}{3}\\\\\Rightarrow s=33.49[/tex]radians
Therefore, the exact arc length is [tex]\frac{32\pi}{3}[/tex] radians.
the arc length in approximate form is 33.49 radians.
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Test 21,753 for divisibility by 2,3,5,9 and 10
Answer:
Step-by-step explanation:
21,753
at unit place=3 not an even number,not equal to 5 and not equal to 0
so 21,753 is not divisible by 2,5 and 10
again
2+1+7+5+3=18 divisible by 3 and 9.
so 21,753 is divisible by 3 and 9.
when price of indomie noodles was lowered from #50 to #40 per unit, quantity demanded increases from 400 to 600 units per week. calculate the coefficient of price elasticity of demand and determine whether by lowering price this firm has made a wise decision
Answer:
The price elasticity of demand is -10
Step-by-step explanation:
Given
[tex]p_1,p_2 = 50,40[/tex]
[tex]q_1,q_2 = 400,500[/tex]
Solving (a): The coefficient of price elasticity of demand (k)
This is calculated as:
[tex]k = \frac{\triangle q}{\triangle p}[/tex]
So, we have:
[tex]k = \frac{500 - 400}{40 - 50}[/tex]
[tex]k = \frac{100}{-10}[/tex]
[tex]k = -10[/tex]
Because |k| > 0, then we can conclude that the company made a wise decision.
An air conditioning system can circulate 310 cubic feet of air per minute. How many cubic yards of air can it circulate per minute? The air conditioning system can circulate about cubic yards of air per minute.
Answer:
310/[tex]3^{3}[/tex] = 310/27 =11.48
Step-by-step explanation:
Answer:
310/ = 310/27 =11.48
Step-by-step explanation:
Tell whether the following two triangles can be proven congruent through SAS.
A.Yes, the two triangles are congruent because they’re both right triangles.
B.Yes, the two triangles are congruent because two sides and their included angle are congruent in both triangles.
C.No, the two triangles can only be proven congruent through SSS.
D.No, the two triangles can only be proven congruent through AAA.
Answer:
C.No, the two triangles can only be proven congruent through SSS.
Question 1 of 10
Estimate the difference of the decimals below by rounding to the nearest
whole number. Enter your answer in the space provided.
46.327
-4.801
Answer:
Step-by-step explanation:
46.327=46 ( neaarest whole number)
-4.801=-5 (nearest whole number)
46-(-5)=46+5=51
log2(6x) – log2 (x)-2
Answer:
xlog(64)−xlog(2)−2
Step-by-step explanation:
Simplify 6log(2) by moving 6 inside the logarithm.
log(2^6)x − log(2)x − 2
Raise 2 to the power of 6.
log(64)x − log(2)x − 2
Reorder factors in log(64)x − log(2)x −2.
HELP ASAP I WILL GIVE BRAINLIST
If sin ∅ = -sqrt{3} OVER 2 and π < ∅ < 3π OVER 2, what are the values of cos ∅ and tan ∅? What is ∅ in degrees and radians? Be sure to show and explain all work.
Step-by-step explanation:
sin ∅ = -(√3)/2
Note that
cos²∅ + sin²∅ = 1
cos²∅ = 1 - sin²∅
= 1 - (-√3 / 2)²
= 1 - (-√3)²/ 2²
= 1 - 3/4
= 1/4
cos²∅ = 1/4
Taking square root of both sides
cos∅ = 1/2
Note that tan∅ = sin∅/cos∅
therefore, tan∅ = -(√3)/2 ÷ 1/2
= -(√3)/2 × 2/1
= -√3
tan∅ = -√3
Since sin∅ = -√3 /2
Then ∅ = -60⁰
The value of ∅ for the given range (third quadrant) is 240⁰.
NB: sin∅ = sin(180-∅)
Also, since 180⁰ is π radians, then ∅ = 4π/3
The sum of the base and height of a triangle is 14 cm. Which of the following equations could be used to find the maximum area of the triangle?
A) A = 0.5x^2 - 15x
B) A = -0.5x^2 + 7x
C) A = -x^2 + 10x
D) A = x^2 - 10x
Answer:
B
Step-by-step explanation:
Let the base of the triangle be b and the height be h.
The sum of the base and height is 14. Thus:
[tex]b+h=14[/tex]
Recall that the area of a triangle is given by:
[tex]\displaystyle A=\frac{1}{2}bh[/tex]
From the first equation, solve for either variable:
[tex]h=14-b[/tex]
Substitute:
[tex]\displaystyle A=\frac{1}{2}b(14-b)[/tex]
Distribute:
[tex]\displaystyle A=\frac{1}{2}(14b-b^2)[/tex]
Distribute:
[tex]\displaystyle A=-0.5b^2+7b[/tex]
Let b = x. Hence:
[tex]A=-0.5x^2+7x[/tex]
Therefore, our answer is B.
What is the value of 3 minus (negative 2)?
A number line going from negative 5 to positive 5.
Answer:
5
Step-by-step explanation:
3-(-2) will become positive 5. so number line will go towards positive 5.
Explain how to divide a decimal by a decimal
Answer:
To divide a decimal by another decimal:
Move the decimal point in the divisor to the right until it is a whole number.
Move the decimal point in the dividend to the right by the same number of places as the decimal point was moved to make the divisor a whole number.
Then divide the new dividend by the new divisor
Step-by-step explanation:
see in the example
11 Emilio makes metal fences.
He is making a fence using this design.
1.44 m
DO NOT WRITE IN THIS AREA
1.8 m
.
The fence will need
3 horizontal metal pieces of length 1.8m
2 tall metal pieces of length 1.44 m
5 medium metal pieces
6 short metal pieces as shown on the diagram.
The heights of the tall, medium and short metal pieces are in the ratio 9:8:7
.
How many metres of metal in total does Emilio need to make the fence?
Answer:
21.4 m
Step-by-step explanation:
Let x represent the sum of the tall metal, medium metal and short metal heights. Since the tall metal has a length of 1.44 m, and the ratio is in 9:8:7, hence:
(9/24) * x = 1.44
x = 3.84 m
For the medium metal pieces:
(8/24) * 3.84 = medium metal height
medium metal height = 1.28 m
For the short metal pieces:
(7/24) * 3.84 = short metal height
short metal height = 1.12 m
Total horizontal metal piece length = 3 * 1.8 m = 5.4 m
Total tall metal piece length = 2 * 1.44 m = 2.88 m
Total medium metal piece length = 5 * 1.28 m = 6.4 m
Total short metal piece length = 6 * 1.12 m = 6.72 m
Total length of metal = 5.4 + 2.88 + 6.4 + 6.72 = 21.4 m
Suppose a large shipment of televisions contained 9% defectives. If a sample of size 393 is selected, what is the probability that the sample proportion will differ from the population proportion by less than 3%
Answer:
0.9624 = 96.24% probability that the sample proportion will differ from the population proportion by less than 3%
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal Probability Distribution
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean [tex]\mu = p[/tex] and standard deviation [tex]s = \sqrt{\frac{p(1-p)}{n}}[/tex]
Suppose a large shipment of televisions contained 9% defectives
This means that [tex]p = 0.09[/tex]
Sample of size 393
This means that [tex]n = 393[/tex]
Mean and standard deviation:
[tex]\mu = p = 0.09[/tex]
[tex]s = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.09*0.91}{393}} = 0.0144[/tex]
What is the probability that the sample proportion will differ from the population proportion by less than 3%?
Proportion between 0.09 - 0.03 = 0.06 and 0.09 + 0.03 = 0.12, which is the p-value of Z when X = 0.12 subtracted by the p-value of Z when X = 0.06.
X = 0.12
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.12 - 0.09}{0.0144}[/tex]
[tex]Z = 2.08[/tex]
[tex]Z = 2.08[/tex] has a p-value of 0.9812
X = 0.06
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.06 - 0.09}{0.0144}[/tex]
[tex]Z = -2.08[/tex]
[tex]Z = -2.08[/tex] has a p-value of 0.0188
0.9812 - 0.0188 = 0.9624
0.9624 = 96.24% probability that the sample proportion will differ from the population proportion by less than 3%
