Question

1. find the surface area and volume of each cone. hint: you will have to use pythagorean theorem to

a) find the missing measurement.

Explanation:

Step1: Find the height of the cone

Using the Pythagorean theorem \(h=\sqrt{l^{2}-r^{2}}\), where \(l = 30\space cm\) (slant height) and \(r=10\space cm\) (radius).

$$h=\sqrt{30^{2}-10^{2}}=\sqrt{900 - 100}=\sqrt{800}=20\sqrt{2}\space cm$$

Step2: Calculate the surface area of the cone

The surface area of a cone \(A=\pi r(r + l)\)
Substitute \(r = 10\space cm\) and \(l=30\space cm\)

$$A=\pi\times10\times(10 + 30)=400\pi\space cm^{2}\approx400\times3.14 = 1256\space cm^{2}$$

Step3: Calculate the volume of the cone

The volume of a cone \(V=\frac{1}{3}\pi r^{2}h\)
Substitute \(r = 10\space cm\) and \(h = 20\sqrt{2}\space cm\)

$$V=\frac{1}{3}\pi\times10^{2}\times20\sqrt{2}=\frac{2000\sqrt{2}\pi}{3}\space cm^{3}\approx\frac{2000\times1.414\times3.14}{3}\approx2966.6\space cm^{3}$$

Answer:

The surface area is approximately \(1256\space cm^{2}\) and the volume is approximately \(2966.6\space cm^{3}\)