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:

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

$$
LATEXBLOCK0
$$

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:

$$ A \approx 201.34\text{ ft}^2 $$

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> \(\text{ft}^2\)