The volume of the solid obtained by rotating the region over the function is given by the integral and V = 44.46 units³
Given data ,
The volume of the solid obtained by rotating the region underneath the graph of the function f(x)=√x² +9 over the interval [0,2] about the y-axis
The formula for Area under definite integral is
∫ₐᵇ f ( x ) = f ( b ) - f ( a )
Since we are rotating the region about the y-axis, the radius is simply the x-coordinate of each point on the curve, or r = x
The height h of each shell is equal to the difference between the y-coordinates of the curve at x and x + Δx, or h = f(x + Δx) - f(x)
Using these expressions for r and h, we can write the volume of each cylindrical shell as:
V(x) = 2πx[f(x + Δx) - f(x)]Δx
V = ∫₀² 2πx[f(x + Δx) - f(x)]dx
As Δx approaches zero, this integral becomes:
V = ∫₀² 2πx√(1 + (f'(x))²) dx
where f'(x) is the derivative of f(x), which is:
f'(x) = x/√(x² + 9)
Substituting this expression for f'(x) into the integral, we get:
V = ∫₀² 2πx√(1 + (x/√(x² + 9))²) dx
This integral can be evaluated using a substitution, u = x² + 9, du/dx = 2x, and dx = du/2x, to get:
V = ∫₉¹³ 2π(x² + 9)^(3/2)/2 dx
V = [4/5 π(x² + 9)^(5/2)]₉¹³
V = 44.46 units³
Hence , the volume of the solid is 44.46 units³
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If f(x) is defined as follows, find (a) f(-3), (b) f(0), and (c) f(4).
x²
if x < 0
if x = 0
3x + 3 ifx>0
f(x) = 0
(a) f(-3)= (Simplify your answer.)
THE
For the given question the values,
f(-1) = 1f(0) = 0f(3) = 13Given value of the function when the condition for x is less than '0' is =
f(x) = x² for x < 0
The value of the function when the condition x is equals to '0' is =
f(x) = 0 for x = 0
The value of the function when the condition x is greater than '0' is =
f(x) = 3x + 4 for x > 0
From the above information,
To find f(-1) we have to use the x value as x². So, f(-1) = (-1)² = 1
To find f(0) we have to use x value as 0. So, f(0) = 0
To find f(3) we have to use the x value as 3x + 4. So, f(3) = 3(3) + 4 = 13.
From the above analysis, we find the values of f(-1), f(0), and f(3).
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The question is done below has square roots and exponents. Pretty easy.
Answer: D. -1
Step-by-step explanation:
Your equation:
[tex]\sqrt[4]{(\sqrt[3]{64}) ^{2} } =(\frac{1}{2} )^{x}[/tex] >We are going to work from the inside first then out
The cube root of 64 is 4 because 4*4*4=64
[tex]\sqrt[4]{(4) ^{2} } =(\frac{1}{2} )^{x}[/tex] > 4² = 4*4=16
[tex]\sqrt[4]{(16) } =(\frac{1}{2} )^{x}[/tex] > the 4th root of 16 is 2 because 2*2*2*2=16
[tex]2 =(\frac{1}{2} )^{x}[/tex] > if you have the same bases you can set the
exponents equal. They are not the same but we
are going to make them the same.
[tex]2^{1} =(\frac{1}{2} )^{x}[/tex] > 2 is the same as 2^1, i can make the bases the
same if I can make the 2 a reciprocal. That
happens when I take the negative exponent of the
number
[tex](\frac{1}{2} )^{-1} =(\frac{1}{2} )^{x}[/tex] >Now that my bases are the same, I can make the
exponents =
-1 = x
Please help!!
If someone could explain how to solve this question I would appreciate it!
2430+15√2=
Answer: The given expression, 2430 + 15√2, cannot be simplified further as it is already in its simplest form. However, you can approximate its value using a calculator or by using a decimal approximation of the square root of 2.
Using a calculator, you can directly evaluate the expression to get:
2430 + 15√2 ≈ 2462.11
Alternatively, you can use a decimal approximation of the square root of 2, which is approximately 1.414:
2430 + 15√2 = 2430 + 15 * 1.414 ≈ 2430 + 21.21 = 2451.21
Therefore, 2430 + 15√2 is approximately equal to 2462.11 or 2451.21, depending on the method used for approximation.
Step-by-step explanation:
What is the numerical expression that represents the sun of eight squared and thirty two
The numerical expression that represents the sum of eight squared and thirty two will be 96.
A numerical expression is a mathematical expression that consists of numbers and mathematical operations such as addition, subtraction, multiplication, division, and exponentiation.
The numerical expression that represents the sum of eight squared and thirty two is
= 8² + 32
= 64 + 32
= 96.
The question is incomplete and the complete question is '' What is the numerical expression that represents the sum of eight squared and thirty two''.
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Help asap!! Please help I don’t get this
The value of arc CD is 110⁰.
The value of arc AD is 120⁰.
What is the measure of the angle?The value of arc CD is calculated by applying intersecting chord theorem, which states that the angle at tangent is half of the arc angle of the two intersecting chords.
angle DEC = ¹/₂ (360 - 2x100) (sum of angle at a point)
angle DEC = ¹/₂ (360 - 200)
angle DEC = 80⁰
The value of arc CD is calculated as follows;
80 = ¹/₂ (CD + 50) (intersecting chord theorem)
2 x 80 = CD + 50
160 = CD + 50
CD = 110⁰
Arc AD = 360 - (50 + 80 + 110) (sum of angles in a circle)
arc AD = 120⁰
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What's the difference between $4 and 36 cents
Answer:
364 cents or $3.64
Step-by-step explanation:
We can first convert $4 into cents. There are a 100 cents in $1, so there are 400 cents in $4. Now we can subtract.
400 - 36 = 364
Difference is 364 cents or $3.64
Question is in image
Answer:
366,699
Step-by-step explanation:
to solve this problem, we can use the following formula for exponential growth:
population = initial population x (1 + growth rate)^time
where the initial population is the current population, the growth rate is the rate of increase per year, and the time is the number of years.
Plugging in the given values, we get:
population = 300,000 x (1 + 0.02)^10
Simplifying, we get:
population = 300,000 x 1.02^10
Using a calculator, we get:
population ≈ 366,698.79
Rounding to the nearest whole number, we get:
population ≈ 366,699
The population in 10 years will be approximately 366,699
Evaluate log_10^3.
a) 100
b) 1, 000
c) 9
d) 3
This is an example of a(n)
Answer:
shape
Step-by-step explanation:
NEED HELP ASAP PLS AND THX PIC IS ATTACHED
Step-by-step explanation:
You would need a calculator in the degree function I believe, but basically for Number 1,
you would set it up in calculator as Sin ^-1 (6/9) or write it down as Sin(X)=(6/9)
2. would be Cos(45)= (X/4) meaning you'd do Cos(45) times 4.
3. is basically Tan(60) =(x/4) so it's basically Tan(60) times 4.
make sure your calculator is in degree mode ? Sorry, I don't remember if you need to be in radiant or degree mode. I can comment or make an edit when I remember.
Eight percent of all college graduates hired by companies stay with the same company for more than five years. The probability, rounded to four decimal places, that in a random sample of 14 such college graduates hired recently by companies, exactly 2 will stay with the same company for more than five years is _?_.
P(X=2) &= {14\choose 2}(0.08)^2(0.92)^{12} \
&= \frac{14!}{2!(14-2)!}(0.08)^2(0.92)^{12} \
[tex]\sf\implies\:&=\frac{14\times13}{2\times1}(0.08)^2(0.92)^{12}[/tex]
&= 91(0.08)^2(0.92)^{12} \
&\approx \boxed{0.2166
[tex]\begin{align}\huge\colorbox{black}{\textcolor{yellow}{\boxed{\sf{I\: hope\: this\: helps !}}}}\end{align}[/tex]
[tex]\begin{align}\colorbox{black}{\textcolor{white}{\underline{\underline{\sf{Please\: mark\: as\: brillinest !}}}}}\end{align}[/tex]
[tex]\textcolor{lime}{\small\textit{If you have any further questions, feel free to ask!}}[/tex]
[tex]\huge{\bigstar{\underline{\boxed{\sf{\color{red}{Sumit\:Roy}}}}}}\\[/tex]
5. The population, P, of a city has grown according to the mathematical model P = 50 000(1.15), where t
is the number of years since 2005.
Using a graphing tool or by hand answer the questions below.
a) What was the population of the town in 2005?
I
b) In what year will the population exceed 100 000?
Answer: Therefore, the population will exceed 100,000 in the year 2005 + 10.73 ≈ 2016.
Step-by-step explanation: a) The population of the town in 2005 is given by the formula, where t = 0 since 2005 is the starting year:
P = 50,000(1.15)^0 = 50,000
Therefore, the population of the town in 2005 was 50,000.
b) We need to find the value of t when the population P exceeds 100,000:
100,000 = 50,000(1.15)^t
Divide both sides by 50,000:
2 = 1.15^t
Take the natural logarithm of both sides:
ln 2 = ln (1.15^t)
Apply the power rule of logarithms:
ln 2 = t ln 1.15
Divide both sides by ln 1.15:
t = ln 2 / ln 1.15
Using a calculator, we get:
t ≈ 10.73
Write an Equation for the ellipse. The center is (1, -3).
The standard form of the equation of the ellipse is (x - 1)² / 16 + (y + 3) / 9 = 1.
How to define the standard form of the equation of an ellipse
In this problem we find the representation of an ellipse on Cartesian plane, whose center is at (h, k) = (1, - 3), whose major semiaxis length is 4 units and whose minor semiaxis length is 3 units. The standard form of the equation of the ellipse is described below:
(x - h)² / a² + (y - k)² / b² = 1
Where:
(h, k) - Coordinates of the center.a - Length of the major semiaxis.b - Length of the minor semiaxis.If we know that (h, k) = (1, - 3), a = 4 and b = 3, then the standard form of the equation of the ellipse is:
(x - 1)² / 16 + (y + 3) / 9 = 1
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You have $12,000 to invest and want to keep your money invested for 8 years. You are considering the following investment options. Choose the investment option that will earn you the most money.
In a case wehereby you have $12,000 to invest and want to keep your money invested for 8 years the investment option that will earn you the most money is c.4.175% compounded annually
What is investment compounded annually?When an investment is compounded annually, it means that the interest earned on the investment is added to the principal amount once a year, and the interest is then calculated on the new total amount for the next year.
For example, if you invest $12,000 at an annual interest rate of 8%, compounded annually, at the end of the first year you will earn the interest of ( $12,000 x 8%) = $960
Then new total amount after one year will be $12,000 + $960 = $12 960 ,
This process will continue for each year of the investment and the formula to calculate the future value (FV) of an investment compounded annually is: FV = P(1 + r)^n
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complete quesation:
You have $12,000 to invest and want to keep your money invested for 8 years. You are considering the following investment options. Choose the investment option that will earn you the most money.
a.
3.99% compounded monthly
b.
4% compounded quarterly
c.
4.175% compounded annually
d.
4.2% simple interest
f(x)=2x³-5x²
g(x)=2x-1
Find (f- g)(x)
Answer:
2x³-5x² - 2x + 1
Step-by-step explanation:
We are given
f(x) = 2x³ - 5x²
g(x) = 2x - 1
and asked to find (f - g)(x)
(f - g)(x) is nothing but f(x) - g(x)
(f- g)(x) = f(x) - g(x) = 2x³-5x² - (2x - 1)
= 2x³-5x² - 2x + 1
3 + 1 + 2 − 7 = + 22
Answer: The answer is false
Answer:
False
Step-by-step explanation:
If you were asking true or false it is false
Select the statement that is true a.16.7-8=2.9×3 b. 4×3.2=17.8-5 c.10.5÷5+1=8.8÷4 d.
Answer:
b
4 x 3.2 = 12.8
17.8 - 5 =12.8
so,
4 x 3.2 = 17.8-5
12.8=12.8
Evaluate (11/16−(3/4)2)×1
Answer:
-166/, -0.82,
Step-by-step explanation:
The above fraction, decimal are all evaluated answer for (11/16−(3/4)2)×1
16
Which graph correctly represents the relationship between arc length and the measure of the corresponding central angle on a circle with radius r?
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Arc
Length
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A graph that correctly represents the relationship between arc length and the measure of the corresponding central angle on a circle with radius r is: C. graph C.
How to calculate the length of the arc?In Mathematics and Geometry, if you want to calculate the length of an arc formed by a circle, you will divide the central angle that is subtended by the arc by 360 degrees and then multiply this fraction by the circumference of the circle.
Mathematically, the length of an arc formed by a circle can be calculated by using the following equation (formula):
Arc length = 2πr × θ/360
In this context, we can reasonably infer and logically deduce that the arc length is directly proportional to the radian measure of the central angle.
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Add 2 1/3 + 4 5/8 writ your answer as a mixed number
Usually, a mixed number is the simplest way to express an improper fraction – but sometimes, the fraction ... Don't express the answer as a decimal. Instead ... So, add the whole number back in to get a final result of 6 1/2. ... Write out the factors for the numerator of your fraction, then write out the factors for the denominator.
Input Signals: P = 0 and Q = 1.
The output of the OR gate will be 1.
What is a NOT Gate?An important component for electronics and computing, the NOT gate or inverter is a basic digital logic gate. It is designed with one input and output that conduct logical negation.
Essentially, this means it turns the input signal to its opposite. When given an input binary value at "1," the method generates "0" as the output and vice versa.
Two input signals, P=0 and Q=1, are subjected to the following process. The message carried by Q is inverted via a NOT gate using its negation feature, returning Q' = 0 at its output.
The resultant value of Q' (evaluated as zero), is then processed using an OR logic operation along with input P into another gate. Outputs from an OR port may only produce "1" if any of the input signal(s) carry a 1. As one of the inputs from this specific procedure provides "0", the result will inevitably be "1".
Consequently, a final analysis reveals that regardless of what the initial value for P was, the result obtained formulating the two signals through a NOT and OR devices matches an outcome of "1".
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I need help, I’m struggling with 3 and 4 can someone help me
Answer:
3 and 4 ==> see work below
[tex]5. \quad\quad f^{-1}(x) = x^{1/7}[/tex]
[tex]6. \quad\quad f^{-1}(x) = -\left(\dfrac{5x}{2}\right)^{1/3}$}\\\text{We can also write this as $-\sqrt[3]{\frac{5x}{2}}$ }\\[/tex]
Step-by-step explanation:
Definition of inverse functions
If f and g are inverse functions, then f(x) = y if and only if g(y) = x
Or, in other words
If f(g(x)) = (g(f(x)) = x
then f and g are inverse functions
Q3
We have f(x) = x + 4 and g(x) = x - 4
To find f(g(x)), substitute g(x) = x - 4 wherever there is an x term in f(x)
f(g(x)) = g(x) + 4
= x - 4 + 4 = x
g(f(x)) = f(x) - 4
= x + 4 - 4 =x
Hence f(x) and g(x) are inverse functions
Q4
[tex]f(x) = \dfrac{1}{4}x^3\\\\g(x) = (4x)^{1/3}[/tex]
[tex]\\\begin{aligned}f(g(x)) &= \dfrac{1}{4} (g(x))^3\\\\\end{aligned}[/tex]
[tex]\begin{aligned}(g(x))^3 &= \left((4x)^{1/3} \right)^3 \\& = (4x)^{\frac{1}{3} \cdot 3}\\& = 4x\end{aligned}[/tex]
Therefore
[tex]\\\begin{aligned}f(g(x)) &= \dfrac{1}{4} (g(x))^3\\&= \dfrac{1}{4} \cdot 4x\\&= x\\\end{aligned}[/tex]
[tex]\begin{aligned}g\left(f(x)\right) & = \left(4f(x)\right)^{1/3}\\&= \left(4 \cdot \dfrac{1}{4}x^3\right)^{1/3}\\& = \left(x^3\right)^{1/3}\\& =x& \end{aligned}[/tex]
So f(x) and g(x) are inverse functions
Q5
[tex]\text{Given $f(x) = x^7 $ we are asked to find inverse $f^{-1}(x)$}[/tex]
[tex]\rm{Let \: y = f(x) = x^7}\\[/tex]
Interchange x and y:
[tex]x = y^7[/tex]
Solve for y:
[tex]y = x^{1/7}[/tex]
The right hand side is the inverse function of f(x)
[tex]f^{-1}(x) = x^{1/7}[/tex]
Q6
[tex]\rm{Given \;f(x) = -\dfrac{2}{5}x^3 \:find\:the\:inverse,\;f^{-1}(x)}[/tex]
Using the same procedure as for Q5
[tex]y=-\dfrac{2}{5}x^3\\\\x=-\dfrac{2}{5}y^3\\\\\text{Solve for y}\\[/tex]
[tex]y^3=-\dfrac{5x}{2}[/tex]
[tex]y=-\left(\dfrac{5x}{2}\right)^{1/3}\\\\\\\text{Inverse of $f(x)$ is $f^{-1}(x) = -\left(\dfrac{5x}{2}\right)^{1/3}$}\\\text{We can also write this as $-\sqrt[3]{\frac{5x}{2}}$ }\\[/tex]
Question is in the image. Please help me solve these
Answer:
Step-by-step explanation:
The model shown below is a perfect cube with a volume of 27 cubic units. Which statement is true about all perfect cubes?
A. A perfect cube represents 3 times the area of a face of the cube.
B. A perfect cube represents the sum of 9 edge lengths of the cube.
C. A perfect cube represents the volume of a cube with equal integer side lengths.
D. A perfect cube represents the surface area of a cube with equal integer side lengths.
The correct statement which is true about all perfect cubes is,
⇒ A perfect cube represents 3 times the area of a face of the cube.
We have to given that;
The model shown below is a perfect cube with a volume of 27 cubic units.
Now, We can formulate;
⇒ V = 27 cubic units.
⇒ V = 3 × 9 cubic units.
⇒ V = 3 × 3² cubic units.
Thus, The correct statement which is true about all perfect cubes is,
⇒ A perfect cube represents 3 times the area of a face of the cube.
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Lara chooses a number less than 20
She divides it by 2 and then adds 6
She then divides this result by 3
Her answer is 4.5
What was the number she started with?
Working backward, the number that Lara started with in performing the mathematical operations was 15.
How the number is determined:We can use mathematical operations to determine the initial number that Lara started with.
Mathematical operations include addition, subtraction, division, multiplication, etc.
Let the number that Lara started with = x
x < 20
x ÷ 2 + 6 = y
y ÷ 3 = 4.5
y = 13.5 (4.5 x 3)
x ÷ 2 + 6 = 13.5
x ÷ 2 = 7.5
x = 15 (7.5 x 2)
Thus, Lara started performing the mathematical operations with 15 and got a final result of 4.5.
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If sun x= 4/5 what is the value of b? 22.5 3b
By following trigonometry identities we get b equals **7**
Define trigonometry identities?Trigonometric identities are equations involving trigonometric functions that hold for all possible values of the variables that occur and for which both sides of the equation are specified. These identities come in use if trigonometric function-based formulas need to be made simpler 1.
There are numerous distinctive trigonometric identities that involve a triangle's side length and angle 2. Only the right-angle triangle 2 is covered by the trigonometric identities. The three main trigonometric functions are sine, cosine, and tangent, while the other three are cotangent, secant, and cosecant.
Some of the most popular trigonometric identities are listed below:
sin²(x) + cos²(x) = 1
- tan(x) = sin(x)/cos(x)
- cot(x) = cos(x)/sin(x)
- sec(x) = 1/cos(x)
- csc(x) = 1/sin(x)
- sin(2x) = 2sin(x)cos(x)
- cos(2x) = cos²(x) - sin²(x)
- tan(2x) = (2tan(x))/(1 - tan²(x))
The use of these identities
.One angle in a right triangle is x°, where sin x°=4/5 . With this knowledge, we can use the inverse sine function (arcsin) to calculate the value of x, which gives us x = arcsin(4/5) = 0.9272952180016122 radians .
In addition, we are informed that NL = 22.5 and NM = 3b. We can get the value of LM, which is equal to√(NL2 + NM2), using the Pythagorean theorem. 2. When the given values are substituted, we obtain LM = √((22.5)2 + (3b)2) = sqrt(506.25 + 9b2).
LM is equivalent to b times cos(x°) since it is the polar opposite of the right angle. Consequently, we can write:
b cos(x°) = √(506.25 + 9b²)
Substituting x = arcsin(4/5), we get:
b cos(arcsin(4/5)) = √(506.25 + 9b²)
Simplifying this equation using trigonometric identities, we get:
b * (√1 - sin²(arcsin(4/5)) = sqrt(506.25 + 9b²)
b × (√(1 - (4/5)²)) = sqrt(506.25 + 9b²)
b× (√(1 - 16/25)) = sqrt(506.25 + 9b²)
b× (√(9/25)) = sqrt(506.25 + 9b²)
3b/5 = √(506.25 + 9b²)
Squaring both sides of the equation, we get:
9b²/25 = 506.25 + 9b²
Solving for b, we get:
b = 7
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If your starting salary is $50,000 and you receive a 4% increase at the end of
every year, what is the total amount, in dollars, you will earn over the first 16
years that you work?
Round your answer to the nearest whole dollar, and express your answer
without using commas.
Answer here
SUBMIT
Answer:
Total amount of becomes after 16 year is $93649 .
Find m/_I. I need help on solving this problem please?
Answer:
∠ I = 60°
Step-by-step explanation:
using the tangent ratio in the right triangle
tan I = [tex]\frac{opposite}{adjacent }[/tex] = [tex]\frac{HJ}{IJ}[/tex] = [tex]\frac{3\sqrt{30} }{3\sqrt{10} }[/tex] = [tex]\frac{\sqrt{30} }{\sqrt{10} }[/tex] = [tex]\sqrt{\frac{30}{10} }[/tex] = [tex]\sqrt{3}[/tex] , then
∠ I = [tex]tan^{-1}[/tex] ([tex]\sqrt{3}[/tex] ) = 60°
Answer:
60°
Step-by-step explanation:
You want the measure of angle I in right triangle HIJ, given that the side opposite is 3√30 and the side adjacent is 3√10.
TangentThe tangent relation is ...
Tan = Opposite/Adjacent
In this triangle, ...
tan(I) = (3√30)/(3√10) = √(30/10) = √3
Then the angle is ...
I = arctan(√3) = 60°
The measure of angle I is 60°.
__
Additional comment
The trig functions of 30° and 45° and their relationships to each other are based on the side relationships of the two "special" right triangles:
30°-60°-90° triangle has side ratios 1 : √3 : 2
45°-45°-90° triangle has side ratios 1 : 1 : √2
Of course, side lengths are in the same order as their opposite angles.
The mnemonic SOH CAH TOA can help you remember the necessary relationships:
Sin = Opposite/HypotenuseCos = Adjacent/HypotenuseTan = Opposite/AdjacentIn 1949, Jackie Robinson hit .342 for the Brooklyn Dodgers; in 1973, Rod Carew hit .350 for the Minnesota Twins. In the 1970s the mean batting average was
.261 and the standard deviation was .0317. Determine which
batting average was more impressive.
In 1973, the batting average of .350 of Rod Carew is more impressive as it has a higher z-score of 2.81
To establish which batting average was more spectacular, we must compare it to the mean and standard deviation of 1970s hitting averages. The case of Jackie Robinson,
z = (x - μ) / σ
z = (.342 - .261) / .0317
z = 2.56
For Rod Carew,
z = (x - μ) / σ
z = (.350 - .261) / .0317
z = 2.81
A higher z-score indicates a more impressive performance relative to the mean. As a result, Rod Carew's .350 batting average was more spectacular, as it had a higher z-score of 2.81 than Jackie Robinson's z-score of 2.56.
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Find the missing dimensions of each figure. Round your answer to the nearest tenth. PLEASE HELP
answer is : r=1.6 in since the volume of cylinder is :
[tex]\pi {r}^{2} h[/tex]