$16$ . A cone has a surface area of $16 \pi \mathrm{m}^{2}$ and a volume of $8 \pi \mathrm{m}^{3}$ . If a similar cone has a surface area of $100 \pi \mathrm{m}^{2}$ . What is the similarity ratio of the smaller cone to the larger cone? What is the volume of the larger cone?

Full solution

16

. A cone has a surface area of

16 \pi \mathrm{m}^{2}

and a volume of

8 \pi \mathrm{m}^{3}

. If a similar cone has a surface area of

100 \pi \mathrm{m}^{2}

. What is the similarity ratio of the smaller cone to the larger cone? What is the volume of the larger cone?

Calculate Scale Factor: Calculate the scale factor for the surface areas of the cones. Given surface area of smaller cone = $16\pi \, \text{m}^2$ , surface area of larger cone = $100\pi \, \text{m}^2$ . Scale factor squared = (Surface area of larger cone) / (Surface area of smaller cone) = $(100\pi) / (16\pi) = 6.25$ . Scale factor = $\sqrt{6.25} = 2.5$ .
Calculate Volume: Calculate the volume of the larger cone using the scale factor. $\newline$ Volume of smaller cone = $8\pi \, \text{m}^3$ . $\newline$ Volume of larger cone = (Scale factor) $^3$ * (Volume of smaller cone) = $2.5^3 * 8\pi = 15.625 * 8\pi = 125\pi \, \text{m}^3$ .