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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
.
.
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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

1 answer

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. \( V = 953.8 , \text{yd}^3, , SA = 551.1 , \text{yd}^2 \).