Question

determine (a) the volume and (b) the surface area of the three-dimensional figure. use the \\(\pi\\) key on your calculator.

a) the volume is
(simplify your answer. type an integer or a decimal rounded to the nearest hundredth as needed.)

b) the surface area is
(simplify your answer. type an integer or a decimal rounded to the nearest hundredth as needed.)

Explanation:

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<reasoning>

Identify the given dimensions of the cone

The figure shows a right circular cone with:

• Diameter \(d = 5\text{ ft}\), which gives a radius \(r = \frac{5}{2} = 2.5\text{ ft}\).
• Height \(h = 23\text{ ft}\).

Calculate the volume of the cone

The volume \(V\) of a cone is given by:
\[
V = \frac{1}{3}\pi r^2 h
\]
Substituting the values:
\[
V = \frac{1}{3}\pi (2.5)^2 (23) = \frac{1}{3}\pi (6.25)(23) = \frac{143.75}{3}\pi \approx 150.5339\text{ ft}^3
\]
Rounding to the nearest hundredth:
\[
V \approx 150.53\text{ ft}^3
\]

Calculate the slant height using the Pythagorean theorem

Using the Pythagorean Theorem knowledge point
\[

$$\begin{aligned} &l = \sqrt{r^2 + h^2}\\ &l = \sqrt{2.5^2 + 23^2} = \sqrt{6.25 + 529} = \sqrt{535.25} \approx 23.13547\text{ ft} \end{aligned}$$

\]

Calculate the total surface area of the cone

The total surface area \(A\) of a cone is the sum of the base area and the lateral area:
\[
A = \pi r^2 + \pi r l = \pi r (r + l)
\]
Substituting the values:
\[
A = \pi (2.5)(2.5 + \sqrt{535.25}) \approx \pi (2.5)(2.5 + 23.13547) = \pi (2.5)(25.63547) \approx 201.3406\text{ ft}^2
\]
Rounding to the nearest hundredth:
\[
A \approx 201.34\text{ ft}^2
\]
</reasoning>

<answer>

Question 1

a) The volume is <blank>150.53</blank> \(\text{ft}^3\)

Question 2

b) The surface area is <blank>201.34</blank> \(\text{ft}^2\)
</answer>

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"Geometry",
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]
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</post_analysis>

Answer:

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<reasoning>

Identify the given dimensions of the cone

The figure shows a right circular cone with:

• Diameter \(d = 5\text{ ft}\), which gives a radius \(r = \frac{5}{2} = 2.5\text{ ft}\).
• Height \(h = 23\text{ ft}\).

Calculate the volume of the cone

The volume \(V\) of a cone is given by:
\[
V = \frac{1}{3}\pi r^2 h
\]
Substituting the values:
\[
V = \frac{1}{3}\pi (2.5)^2 (23) = \frac{1}{3}\pi (6.25)(23) = \frac{143.75}{3}\pi \approx 150.5339\text{ ft}^3
\]
Rounding to the nearest hundredth:
\[
V \approx 150.53\text{ ft}^3
\]

Calculate the slant height using the Pythagorean theorem

Using the Pythagorean Theorem knowledge point
\[

$$\begin{aligned} &l = \sqrt{r^2 + h^2}\\ &l = \sqrt{2.5^2 + 23^2} = \sqrt{6.25 + 529} = \sqrt{535.25} \approx 23.13547\text{ ft} \end{aligned}$$

\]

Calculate the total surface area of the cone

The total surface area \(A\) of a cone is the sum of the base area and the lateral area:
\[
A = \pi r^2 + \pi r l = \pi r (r + l)
\]
Substituting the values:
\[
A = \pi (2.5)(2.5 + \sqrt{535.25}) \approx \pi (2.5)(2.5 + 23.13547) = \pi (2.5)(25.63547) \approx 201.3406\text{ ft}^2
\]
Rounding to the nearest hundredth:
\[
A \approx 201.34\text{ ft}^2
\]
</reasoning>

<answer>

Question 1

a) The volume is <blank>150.53</blank> \(\text{ft}^3\)

Question 2

b) The surface area is <blank>201.34</blank> \(\text{ft}^2\)
</answer>

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"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Geometry",
"Volume of Cone"
]
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