+2441 8) This one will require the formula that yields the volume of a cylinder. \(V_{\text{cylinder}}=\pi r^2h\). We can manipulate this formula so that we can find any missing information such as the height, in this case.
| \(V_{\text{cylinder}}=\pi r^2h\) | We know what the volume is, and we know the height, so finding the radius is simply a matter of isolating the variable. | ||
| \(27143=15\pi r^2\) | Divide by 15 pi first. | ||
| \(\frac{27143}{15\pi}=r^2\) | Take the square root of both sides. | ||
| \(|r|=\sqrt{\frac{27143}{15\pi}}\) | The absolute value splits the answer into two possibilities. | ||
| In the context of geometry, negative side lengths are nonsensical, so let's just reject the answer now. | ||
| \(r=\sqrt{\frac{27143}{15\pi}}\approx 24\text{m}\) | The radius is a one-dimensional part of a cylinder, so the units should be in one dimension, too. | ||
9) If the height of the un-consumed soup was 8 centimeters tall and 3-centimeters-worth of soup is consumed, then 5 centimeters of soup remains. We already know the radius of this soup can (that I assume is cylinder-shaped despite not being explicitly stated), so we can determine the volume.
| \(V_{\text{cylinder}}=\pi r^2h\) | Plug in the known values. |
| \(V_{\text{cylinder}}=\pi*12^2*5\) | Now, combine like terms. |
| \(V_{\text{cylinder}}=720\pi\approx 2262\text{cm}^3\) | Volume is always expressed as a cubic unit. |
10) If the town park enlarges its area by a factor of 5, then both dimensions of the park are affected by this scale factor. For example, if we assume, for the sake of understanding, that the park is perfectly rectangular with dimensions 3yd by 100yd, then both dimensions (the length and the width) would be affected by this scale factor. This means that, on area, the scale factor actually affects the area by its square, or 25 in this case.
\(300\text{yd}^2*25=7500\text{yd}^2\)
+2441 8) This one will require the formula that yields the volume of a cylinder. \(V_{\text{cylinder}}=\pi r^2h\). We can manipulate this formula so that we can find any missing information such as the height, in this case.
| \(V_{\text{cylinder}}=\pi r^2h\) | We know what the volume is, and we know the height, so finding the radius is simply a matter of isolating the variable. | ||
| \(27143=15\pi r^2\) | Divide by 15 pi first. | ||
| \(\frac{27143}{15\pi}=r^2\) | Take the square root of both sides. | ||
| \(|r|=\sqrt{\frac{27143}{15\pi}}\) | The absolute value splits the answer into two possibilities. | ||
| In the context of geometry, negative side lengths are nonsensical, so let's just reject the answer now. | ||
| \(r=\sqrt{\frac{27143}{15\pi}}\approx 24\text{m}\) | The radius is a one-dimensional part of a cylinder, so the units should be in one dimension, too. | ||
9) If the height of the un-consumed soup was 8 centimeters tall and 3-centimeters-worth of soup is consumed, then 5 centimeters of soup remains. We already know the radius of this soup can (that I assume is cylinder-shaped despite not being explicitly stated), so we can determine the volume.
| \(V_{\text{cylinder}}=\pi r^2h\) | Plug in the known values. |
| \(V_{\text{cylinder}}=\pi*12^2*5\) | Now, combine like terms. |
| \(V_{\text{cylinder}}=720\pi\approx 2262\text{cm}^3\) | Volume is always expressed as a cubic unit. |
10) If the town park enlarges its area by a factor of 5, then both dimensions of the park are affected by this scale factor. For example, if we assume, for the sake of understanding, that the park is perfectly rectangular with dimensions 3yd by 100yd, then both dimensions (the length and the width) would be affected by this scale factor. This means that, on area, the scale factor actually affects the area by its square, or 25 in this case.
\(300\text{yd}^2*25=7500\text{yd}^2\)
