finding volume
example 3 find the volume of a rectangular prism with a length of 10 centimeters, a width of 4 centimeters, and a height of 5 centimeters.
volume = lwh
= (10)(4)(5)
= 200
the volume is 200 cubic centimeters.
find the volume of the solid.
7. cube with edge length 4 inches
8. sphere with radius 2 feet
9. rectangular prism with length 4 feet, width 2 feet, and height 6 feet
10. cylinder with radius 3 centimeters and height 5 centimeters
11. persevere the edge length of a cube increases from a inches to ka inches, where k 0. by how much does the volume increase?

Question

finding volume
example 3 find the volume of a rectangular prism with a length of 10 centimeters, a width of 4 centimeters, and a height of 5 centimeters.
volume = lwh
= (10)(4)(5)
= 200
the volume is 200 cubic centimeters.
find the volume of the solid.
7. cube with edge length 4 inches
8. sphere with radius 2 feet
9. rectangular prism with length 4 feet, width 2 feet, and height 6 feet
10. cylinder with radius 3 centimeters and height 5 centimeters
11. persevere the edge length of a cube increases from a inches to ka inches, where k > 0. by how much does the volume increase?

Accepted Answer

### Problem 7: Cube with edge length 4 inches
# Explanation:
## Step1: Recall cube volume formula
The volume $ V $ of a cube with edge length $ s $ is given by $ V = s^3 $.
## Step2: Substitute $ s = 4 $
Substitute $ s = 4 $ into the formula: $ V = 4^3 $.
## Step3: Calculate the volume
$ 4^3 = 4\times4\times4 = 64 $.
# Answer:
The volume of the cube is 64 cubic inches.

### Problem 8: Sphere with radius 2 feet
# Explanation:
## Step1: Recall sphere volume formula
The volume $ V $ of a sphere with radius $ r $ is given by $ V=\frac{4}{3}\pi r^3 $.
## Step2: Substitute $ r = 2 $
Substitute $ r = 2 $ into the formula: $ V=\frac{4}{3}\pi(2)^3 $.
## Step3: Calculate the volume
First, calculate $ 2^3=8 $. Then, $ V = \frac{4}{3}\pi\times8=\frac{32}{3}\pi\approx33.51 $ (if we use $ \pi\approx3.14 $, $ \frac{32}{3}\times3.14\approx33.51 $).
# Answer:
The volume of the sphere is $ \frac{32}{3}\pi $ cubic feet (or approximately 33.51 cubic feet).

### Problem 9: Rectangular prism with length 4 feet, width 2 feet, and height 6 feet
# Explanation:
## Step1: Recall rectangular prism volume formula
The volume $ V $ of a rectangular prism with length $ l $, width $ w $, and height $ h $ is $ V = lwh $.
## Step2: Substitute $ l = 4 $, $ w = 2 $, $ h = 6 $
Substitute the values into the formula: $ V=(4)(2)(6) $.
## Step3: Calculate the volume
Multiply the numbers: $ 4\times2\times6 = 48 $.
# Answer:
The volume of the rectangular prism is 48 cubic feet.

### Problem 10: Cylinder with radius 3 centimeters and height 5 centimeters
# Explanation:
## Step1: Recall cylinder volume formula
The volume $ V $ of a cylinder with radius $ r $ and height $ h $ is $ V=\pi r^2h $.
## Step2: Substitute $ r = 3 $, $ h = 5 $
Substitute the values into the formula: $ V=\pi(3)^2(5) $.
## Step3: Calculate the volume
First, calculate $ 3^2 = 9 $. Then, $ V=\pi\times9\times5 = 45\pi\approx141.37 $ (if we use $ \pi\approx3.14 $, $ 45\times3.14 = 141.3 $).
# Answer:
The volume of the cylinder is $ 45\pi $ cubic centimeters (or approximately 141.3 cubic centimeters).

### Problem 11: Edge length of a cube increases from $ a $ inches to $ ka $ inches ($ k>0 $)
# Explanation:
## Step1: Find initial volume
The initial volume $ V_1 $ of the cube with edge length $ a $ is $ V_1=a^3 $.
## Step2: Find final volume
The final volume $ V_2 $ of the cube with edge length $ ka $ is $ V_2=(ka)^3=k^3a^3 $ (using the property $ (ab)^n=a^nb^n $).
## Step3: Find the increase in volume
The increase in volume $ \Delta V $ is $ V_2 - V_1 $. Substitute $ V_1 $ and $ V_2 $: $ \Delta V=k^3a^3 - a^3=a^3(k^3 - 1) $.
# Answer:
The volume increases by $ a^3(k^3 - 1) $ cubic inches.

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QUESTION IMAGE

Question

Explanation:

Problem 7: Cube with edge length 4 inches

Step1: Recall cube volume formula

Step2: Substitute \( s = 4 \)

Step3: Calculate the volume

Step1: Recall sphere volume formula

Step2: Substitute \( r = 2 \)

Step3: Calculate the volume

Step1: Recall rectangular prism volume formula

Step2: Substitute \( l = 4 \), \( w = 2 \), \( h = 6 \)

Step3: Calculate the volume

Answer:

Problem 8: Sphere with radius 2 feet