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Wilber wants to make the pond in his backyard bigger. It is currently a cone with a radius of 30 meters and a depth of 15 meters. He wants to increase the radius to 40 meters and the depth to 20 meters. How much more water will the pond hold, in thousands of cubic meters?
(Round to the nearest thousand cubic meters.)

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Wilber wants to make the pond in his backyard bigger. It is currently a cone with a radius of $30$ meters and a depth of $15$ meters. He wants to increase the radius to $40$ meters and the depth to $20$ meters. How much more water will the pond hold, in thousands of cubic meters? $\newline$ (Round to the nearest thousand cubic meters.) $\newline$ $\square$

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Circles: word problems

Full solution

Q. Wilber wants to make the pond in his backyard bigger. It is currently a cone with a radius of

30

meters and a depth of

15

meters. He wants to increase the radius to

40

meters and the depth to

20

meters. How much more water will the pond hold, in thousands of cubic meters?

\newline

(Round to the nearest thousand cubic meters.)

\newline

\square

Calculate Volume Original Cone: Calculate the volume of the original cone-shaped pond. The formula for the volume of a cone is $V = \frac{1}{3}\pi r^2 h$ , where $r$ is the radius and $h$ is the height (or depth in this case). Original volume $V_{\text{original}}$ = $\frac{1}{3}\pi(30 \text{ meters})^2(15 \text{ meters})$
Calculate Original Volume: Perform the calculation for the original volume. $\newline$ $V_{\text{original}} = \frac{1}{3}\pi(900)(15)$ $\newline$ $V_{\text{original}} = \pi(300)(15)$ $\newline$ $V_{\text{original}} = 4500\pi$ cubic meters
Calculate Volume Expanded Cone: Calculate the volume of the expanded cone-shaped pond. Expanded volume $V_{\text{expanded}}$ = $(1/3)\pi(40 \text{ meters})^2(20 \text{ meters})$
Calculate Expanded Volume: Perform the calculation for the expanded volume. $\newline$ $V_{\text{expanded}} = \frac{1}{3}\pi(1600)(20)$ $\newline$ $V_{\text{expanded}} = \pi(533.33)(20)$ $\newline$ $V_{\text{expanded}} = 10666.67\pi$ cubic meters
Calculate Difference in Volume: Calculate the difference in volume to find out how much more water the expanded pond will hold. $\newline$ Difference in volume ( $\Delta V$ ) = $V_{\text{expanded}} - V_{\text{original}}$ $\newline$ $\Delta V = 10666.67\pi - 4500\pi$
Calculate Difference in Thousands: Perform the calculation for the difference in volume. $\newline$ $\Delta V = (10666.67 - 4500)\pi$ $\newline$ $\Delta V = 6166.67\pi$ cubic meters
Convert to Nearest Thousand: Convert the difference in volume to thousands of cubic meters and round to the nearest thousand. $\newline$ $\Delta V$ (in thousands of cubic meters) = $\frac{6166.67\pi}{1000}$ $\newline$ Since $\pi$ is approximately $3$ . $14159$ , we calculate: $\newline$ $\Delta V \approx \frac{6166.67 \times 3.14159}{1000}$ $\newline$ $\Delta V \approx 19.374$ cubic kilometers $\newline$ Rounded to the nearest thousand cubic meters: $\Delta V \approx 19,000$ cubic meters

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