Question
Find the volume and surface area of the figure. Use 3.14 as the approximate value of pi. Round your answer to the nearest tenth, if necessary.
nbsp 9 yd
nbsp 15 yd
.
.
.
Question content area right
Part 1
A.
V = 3 comma 815.1 yd3 SA = 975 yd squared
B.
V = 212 yd3 SA = 1 comma 356.5 yd squared
C.
V = 423.9 yd3 SA = 275.5 yd squared
D.
V = 953.8 yd3 SA = 551.1 yd squared
nbsp 9 yd
nbsp 15 yd
.
.
.
Question content area right
Part 1
A.
V = 3 comma 815.1 yd3 SA = 975 yd squared
B.
V = 212 yd3 SA = 1 comma 356.5 yd squared
C.
V = 423.9 yd3 SA = 275.5 yd squared
D.
V = 953.8 yd3 SA = 551.1 yd squared
Answers
GPT-4o mini
To determine the volume and surface area of the figure, we first need to identify the shape in question. The dimensions given (9 yd and 15 yd) could suggest a cylinder or a rectangular prism. However, since specific figures aren't described, we'll assume it refers to a common geometric figure with these dimensions.
Assuming this refers to a cylinder where:
- The height (h) = 15 yd
- The diameter = 9 yd (which gives a radius r = 4.5 yd)
### Volume of a Cylinder
The formula for the volume (V) of a cylinder is:
\[ V = \pi r^2 h \]
Using the provided value of \(\pi \approx 3.14\):
1. Calculate the radius squared:
\[ r^2 = (4.5 \text{ yd})^2 = 20.25 \text{ yd}^2 \]
2. Calculate the volume:
\[ V = 3.14 \times 20.25 \text{ yd}^2 \times 15 \text{ yd} \]
\[ V = 3.14 \times 20.25 \times 15 \]
\[ V \approx 3.14 \times 303.75 \]
\[ V \approx 953.625 \text{ yd}^3 \]
\[ V \approx 953.6 \text{ yd}^3 \] (rounded to the nearest tenth)
### Surface Area of a Cylinder
The formula for the surface area (SA) of a cylinder is:
\[ SA = 2\pi rh + 2\pi r^2 \]
1. Calculate the lateral surface area (2πrh):
\[ Lateral\,SA = 2 \times 3.14 \times 4.5 \text{ yd} \times 15 \text{ yd} \]
\[ Lateral\,SA = 2 \times 3.14 \times 67.5 \text{ yd}^2 \]
\[ Lateral\,SA \approx 2 \times 212.65 \text{ yd}^2 \]
\[ Lateral\,SA \approx 425.3 \text{ yd}^2 \]
2. Calculate the area of the top and bottom (2πr²):
\[ Base\,SA = 2 \times 3.14 \times (4.5 \text{ yd})^2 \]
\[ Base\,SA = 2 \times 3.14 \times 20.25 \text{ yd}^2 \]
\[ Base\,SA \approx 2 \times 63.94 \text{ yd}^2 \]
\[ Base\,SA \approx 127.88 \text{ yd}^2 \]
3. Total surface area:
\[ SA = Lateral\,SA + Base\,SA \]
\[ SA \approx 425.3 \text{ yd}^2 + 127.88 \text{ yd}^2 \]
\[ SA \approx 553.18 \text{ yd}^2 \]
\[ SA \approx 553.2 \text{ yd}^2 \] (rounded to the nearest tenth)
Based on our calculations for the cylinder:
- Volume \( V \approx 953.6 \, \text{yd}^3 \)
- Surface Area \( SA \approx 553.2 \, \text{yd}^2 \)
Looking at the options:
- D. \( V = 953.8 \, \text{yd}^3, \, SA = 551.1 \, \text{yd}^2 \)
This option is the closest approximation to our calculations.
So, the final answer is:
**D. \( V = 953.8 \, \text{yd}^3, \, SA = 551.1 \, \text{yd}^2 \)**.
Assuming this refers to a cylinder where:
- The height (h) = 15 yd
- The diameter = 9 yd (which gives a radius r = 4.5 yd)
### Volume of a Cylinder
The formula for the volume (V) of a cylinder is:
\[ V = \pi r^2 h \]
Using the provided value of \(\pi \approx 3.14\):
1. Calculate the radius squared:
\[ r^2 = (4.5 \text{ yd})^2 = 20.25 \text{ yd}^2 \]
2. Calculate the volume:
\[ V = 3.14 \times 20.25 \text{ yd}^2 \times 15 \text{ yd} \]
\[ V = 3.14 \times 20.25 \times 15 \]
\[ V \approx 3.14 \times 303.75 \]
\[ V \approx 953.625 \text{ yd}^3 \]
\[ V \approx 953.6 \text{ yd}^3 \] (rounded to the nearest tenth)
### Surface Area of a Cylinder
The formula for the surface area (SA) of a cylinder is:
\[ SA = 2\pi rh + 2\pi r^2 \]
1. Calculate the lateral surface area (2πrh):
\[ Lateral\,SA = 2 \times 3.14 \times 4.5 \text{ yd} \times 15 \text{ yd} \]
\[ Lateral\,SA = 2 \times 3.14 \times 67.5 \text{ yd}^2 \]
\[ Lateral\,SA \approx 2 \times 212.65 \text{ yd}^2 \]
\[ Lateral\,SA \approx 425.3 \text{ yd}^2 \]
2. Calculate the area of the top and bottom (2πr²):
\[ Base\,SA = 2 \times 3.14 \times (4.5 \text{ yd})^2 \]
\[ Base\,SA = 2 \times 3.14 \times 20.25 \text{ yd}^2 \]
\[ Base\,SA \approx 2 \times 63.94 \text{ yd}^2 \]
\[ Base\,SA \approx 127.88 \text{ yd}^2 \]
3. Total surface area:
\[ SA = Lateral\,SA + Base\,SA \]
\[ SA \approx 425.3 \text{ yd}^2 + 127.88 \text{ yd}^2 \]
\[ SA \approx 553.18 \text{ yd}^2 \]
\[ SA \approx 553.2 \text{ yd}^2 \] (rounded to the nearest tenth)
Based on our calculations for the cylinder:
- Volume \( V \approx 953.6 \, \text{yd}^3 \)
- Surface Area \( SA \approx 553.2 \, \text{yd}^2 \)
Looking at the options:
- D. \( V = 953.8 \, \text{yd}^3, \, SA = 551.1 \, \text{yd}^2 \)
This option is the closest approximation to our calculations.
So, the final answer is:
**D. \( V = 953.8 \, \text{yd}^3, \, SA = 551.1 \, \text{yd}^2 \)**.