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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.) ◻

Wilber wants to make the pond in his backyard bigger. It is currently a cone with a radius of 3030 meters and a depth of 1515 meters. He wants to increase the radius to 4040 meters and the depth to 2020 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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Q. Wilber wants to make the pond in his backyard bigger. It is currently a cone with a radius of 3030 meters and a depth of 1515 meters. He wants to increase the radius to 4040 meters and the depth to 2020 meters. How much more water will the pond hold, in thousands of cubic meters? \newline(Round to the nearest thousand cubic meters.)\newline□\square
  1. Calculate Volume Original Cone: Calculate the volume of the original cone-shaped pond. The formula for the volume of a cone is V=13πr2hV = \frac{1}{3}\pi r^2 h, where rr is the radius and hh is the height (or depth in this case). Original volume VoriginalV_{\text{original}} = 13π(30 meters)2(15 meters)\frac{1}{3}\pi(30 \text{ meters})^2(15 \text{ meters})
  2. Calculate Original Volume: Perform the calculation for the original volume.\newlineVoriginal=13π(900)(15)V_{\text{original}} = \frac{1}{3}\pi(900)(15)\newlineVoriginal=π(300)(15)V_{\text{original}} = \pi(300)(15)\newlineVoriginal=4500πV_{\text{original}} = 4500\pi cubic meters
  3. Calculate Volume Expanded Cone: Calculate the volume of the expanded cone-shaped pond. Expanded volume VexpandedV_{\text{expanded}} = (1/3)π(40 meters)2(20 meters)(1/3)\pi(40 \text{ meters})^2(20 \text{ meters})
  4. Calculate Expanded Volume: Perform the calculation for the expanded volume.\newlineVexpanded=13π(1600)(20)V_{\text{expanded}} = \frac{1}{3}\pi(1600)(20)\newlineVexpanded=π(533.33)(20)V_{\text{expanded}} = \pi(533.33)(20)\newlineVexpanded=10666.67πV_{\text{expanded}} = 10666.67\pi cubic meters
  5. Calculate Difference in Volume: Calculate the difference in volume to find out how much more water the expanded pond will hold.\newlineDifference in volume (ΔV\Delta V) = Vexpanded−VoriginalV_{\text{expanded}} - V_{\text{original}}\newlineΔV=10666.67π−4500π\Delta V = 10666.67\pi - 4500\pi
  6. Calculate Difference in Thousands: Perform the calculation for the difference in volume.\newlineΔV=(10666.67−4500)π\Delta V = (10666.67 - 4500)\pi\newlineΔV=6166.67π\Delta V = 6166.67\pi cubic meters
  7. Convert to Nearest Thousand: Convert the difference in volume to thousands of cubic meters and round to the nearest thousand.\newlineΔV\Delta V (in thousands of cubic meters) = 6166.67π1000\frac{6166.67\pi}{1000}\newlineSince π\pi is approximately 33.1415914159, we calculate:\newlineΔV≈6166.67×3.141591000\Delta V \approx \frac{6166.67 \times 3.14159}{1000}\newlineΔV≈19.374\Delta V \approx 19.374 cubic kilometers\newlineRounded to the nearest thousand cubic meters: ΔV≈19,000\Delta V \approx 19,000 cubic meters

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