Question

A reservoir is in the shape of a frustum of a right circular cone. It is 8 m across at the top and 4 m cross at the bottom. It is 6 m deep then its capacity is

• A. \(176\ m^3\)
• B. \(174\ m^3\)
• C. \(127\ m^3\)
• D. \(170\ m^3\)

Answer 1

The Correct Option is A

To solve the problem, we need to determine the capacity (volume) of a reservoir shaped like a frustum of a right circular cone. The given dimensions are:

Diameter at the top: \(8 \, \text{m}\)
Diameter at the bottom: \(4 \, \text{m}\)
Depth (height) of the frustum: \(6 \, \text{m}\)

Step 1: Formula for the Volume of a Frustum of a Cone The volume \(V\) of a frustum of a cone is given by: \[ V = \frac{1}{3} \pi h \left( R^2 + Rr + r^2 \right) \] where:
\(R\) is the radius of the larger base (top),
\(r\) is the radius of the smaller base (bottom),
\(h\) is the height (depth) of the frustum.

Step 2: Calculate the Radii The radii are half of the respective diameters: - Radius at the top (\(R\)): \[ R = \frac{\text{Diameter at the top}}{2} = \frac{8}{2} = 4 \, \text{m} \] - Radius at the bottom (\(r\)): \[ r = \frac{\text{Diameter at the bottom}}{2} = \frac{4}{2} = 2 \, \text{m} \] Step 3: Substitute Known Values into the Volume Formula We are given: - \(R = 4 \, \text{m}\), - \(r = 2 \, \text{m}\), - \(h = 6 \, \text{m}\). Substitute these values into the formula: \[ V = \frac{1}{3} \pi h \left( R^2 + Rr + r^2 \right) \] \[ V = \frac{1}{3} \pi (6) \left( 4^2 + 4 \cdot 2 + 2^2 \right) \] Step 4: Simplify the Expression First, calculate the terms inside the parentheses: \[ 4^2 = 16, \quad 4 \cdot 2 = 8, \quad 2^2 = 4 \] \[ R^2 + Rr + r^2 = 16 + 8 + 4 = 28 \] Now substitute back: \[ V = \frac{1}{3} \pi (6) (28) \] \[ V = \frac{1}{3} \cdot 6 \cdot 28 \cdot \pi \] \[ V = 2 \cdot 28 \cdot \pi \] \[ V = 56 \pi \] Step 5: Approximate the Volume Using \(\pi \approx 3.14\): \[ V \approx 56 \cdot 3.14 = 175.84 \, \text{m}^3 \] Step 6: Match with the Given Options The closest value to \(175.84 \, \text{m}^3\) among the options is \(176 \, \text{m}^3\).

Final Answer: \[ {176 \, \text{m}^3} \]