Question
A highway traffic cone consists of a cone centered on a square prism base. The cone has a height of 36 in. and a diameter of 10 in. The sides of the base are 14.5 in., and the height is 1.5 in. If the traffic cone was solid, what would be its volume? Round your answer to the nearest whole number.(1 point)Responses942 in.3 942 inches cubed3,143 in.3 3,143 inches cubed1,258 in.31,258 inches cubed615 in.3
Answers
GPT-4o mini
To find the volume of the highway traffic cone, we first need to calculate the volume of the conical part and then add the volume of the square prism base.
**Step 1: Volume of the cone**
The formula for the volume \( V \) of a cone is:
\[
V = \frac{1}{3} \pi r^2 h
\]
where:
- \( r \) is the radius of the base of the cone,
- \( h \) is the height of the cone.
Given data:
- Diameter of the cone = 10 in, so radius \( r = \frac{10}{2} = 5 \) in.
- Height of the cone \( h = 36 \) in.
Substituting the values into the formula:
\[
V = \frac{1}{3} \pi (5)^2 (36)
\]
\[
V = \frac{1}{3} \pi (25)(36)
\]
\[
V = \frac{1}{3} \pi (900)
\]
\[
V = 300\pi \text{ in}^3
\]
Using \( \pi \approx 3.14 \):
\[
V \approx 300 \times 3.14 \approx 942 \text{ in}^3
\]
**Step 2: Volume of the square prism base**
The formula for the volume \( V \) of a square prism is:
\[
V = \text{Base Area} \times \text{Height}
\]
Given:
- Side length of the square base = 14.5 in,
- Height of the prism \( h = 1.5 \) in.
Calculating the base area:
\[
\text{Base Area} = (14.5)^2 = 210.25 \text{ in}^2
\]
Now calculating the volume of the prism:
\[
V = 210.25 \text{ in}^2 \times 1.5 \text{ in} = 315.375 \text{ in}^3
\]
**Step 3: Total Volume**
Now we add the volumes of the cone and the square prism:
\[
\text{Total Volume} = 942 \text{ in}^3 + 315.375 \text{ in}^3 \approx 1257.375 \text{ in}^3
\]
Rounding to the nearest whole number, the total volume is:
\[
\boxed{1258} \text{ in}^3
\]
**Step 1: Volume of the cone**
The formula for the volume \( V \) of a cone is:
\[
V = \frac{1}{3} \pi r^2 h
\]
where:
- \( r \) is the radius of the base of the cone,
- \( h \) is the height of the cone.
Given data:
- Diameter of the cone = 10 in, so radius \( r = \frac{10}{2} = 5 \) in.
- Height of the cone \( h = 36 \) in.
Substituting the values into the formula:
\[
V = \frac{1}{3} \pi (5)^2 (36)
\]
\[
V = \frac{1}{3} \pi (25)(36)
\]
\[
V = \frac{1}{3} \pi (900)
\]
\[
V = 300\pi \text{ in}^3
\]
Using \( \pi \approx 3.14 \):
\[
V \approx 300 \times 3.14 \approx 942 \text{ in}^3
\]
**Step 2: Volume of the square prism base**
The formula for the volume \( V \) of a square prism is:
\[
V = \text{Base Area} \times \text{Height}
\]
Given:
- Side length of the square base = 14.5 in,
- Height of the prism \( h = 1.5 \) in.
Calculating the base area:
\[
\text{Base Area} = (14.5)^2 = 210.25 \text{ in}^2
\]
Now calculating the volume of the prism:
\[
V = 210.25 \text{ in}^2 \times 1.5 \text{ in} = 315.375 \text{ in}^3
\]
**Step 3: Total Volume**
Now we add the volumes of the cone and the square prism:
\[
\text{Total Volume} = 942 \text{ in}^3 + 315.375 \text{ in}^3 \approx 1257.375 \text{ in}^3
\]
Rounding to the nearest whole number, the total volume is:
\[
\boxed{1258} \text{ in}^3
\]