Let's solve each of the questions step by step.
Question 1: We know the formula for the volume of a cone: \[ V = \frac{1}{3} \pi r^2 h \] Given that \( V = 100\pi \) cubic inches and \( h = 12 \) inches, we can plug in these values: \[ 100\pi = \frac{1}{3} \pi r^2 (12) \] We can simplify this by cancelling \( \pi \): \[ 100 = \frac{1}{3} r^2 (12) \] Now, multiply both sides by 3: \[ 300 = r^2 (12) \] Now, divide both sides by 12: \[ 25 = r^2 \] Taking the square root of both sides gives: \[ r = 5 \text{ inches} \]
The radius of the cone is 5 inches.
Question 2: To find the volume of a cone, Nevaeh must know that the volume of a cone is one-third of the volume of the cylinder with the same height and base radius. Therefore: To find the volume of the cone, Nevaeh must multiply the volume of the cylinder by 1/3.
Question 3: The formula for the volume of a cone again is: \[ V = \frac{1}{3} \pi r^2 h \] Given \( V = 225\pi \) cubic meters and the diameter is \( 10 \) meters, the radius \( r = \frac{10}{2} = 5 \) meters. Plugging in the values we have: \[ 225\pi = \frac{1}{3} \pi (5^2) h \] Simplifying this gives: \[ 225 = \frac{1}{3} \cdot 25 h \] Multiplying both sides by 3: \[ 675 = 25h \] Now, divide by 25: \[ h = 27 \text{ meters} \]
The height of the cone is 27 meters.
Question 4: Again using the formula for the volume of a cone: \[ V = \frac{1}{3} \pi r^2 h \] Given \( r = 12 \) feet and \( h = \frac{1}{2}(12) = 6 \) feet, we find: \[ V = \frac{1}{3} \pi (12^2)(6) \] Calculating \( 12^2 = 144 \), we have: \[ V = \frac{1}{3} \cdot 3.14 \cdot 144 \cdot 6 \] Calculating this step by step: \[ = \frac{1}{3} \cdot 3.14 \cdot 864 \] Calculating \( 3.14 \cdot 864 \approx 2714.56 \): \[ V \approx \frac{2714.56}{3} \approx 904.8533333 \approx 904.9 \text{ cubic feet (rounded to the nearest tenth)} \]
The volume of Erik’s cone is approximately 904.9 cubic feet.
Question 5: Using the volume formula for the cone given \( r = 6 \) feet and \( h = 8 \) feet: \[ V = \frac{1}{3} \pi r^2 h \] Plugging in the values: \[ V = \frac{1}{3} \cdot 3.14 \cdot (6^2) \cdot 8 \] Calculating \( 6^2 = 36 \): \[ V = \frac{1}{3} \cdot 3.14 \cdot 36 \cdot 8 \] Calculating \( 36 \cdot 8 = 288 \): \[ V = \frac{1}{3} \cdot 3.14 \cdot 288 \] Calculating \( 3.14 \cdot 288 = 904.32 \): \[ V = \frac{904.32}{3} \approx 300.144 \approx 300 \text{ cubic feet (rounded to the nearest cubic foot)} \]
The volume of the cone is approximately 300 cubic feet.
So to summarize:
- Radius: 5 inches
- To find the volume of the cone, multiply by 1/3.
- Height: 27 meters
- Volume of Erik's cone: 904.9 cubic feet
- Volume of the cone in the image: 300 cubic feet.