Math

QuestionA farmer has a tank shaped like a cylinder with a cone on top. Cone radius is 10 m10 \mathrm{~m}, height is 2 m2 \mathrm{~m}; cylinder height is 5 m5 \mathrm{~m}. Find the volumes of the cone and cylinder.

Studdy Solution

STEP 1

Assumptions1. The base of the cone has a radius of10 m. The perpendicular height of the cone is m3. The height of the cylinder is5 m4. The radius of the base of the cylinder is the same as the radius of the base of the cone (10 m), as it is stated that the tank is a cylinder topped with a cone5. The volume of a cone is given by the formula Vcone=13πrhV_{cone} = \frac{1}{3} \pi r^ h
6. The volume of a cylinder is given by the formula Vcylinder=πrhV_{cylinder} = \pi r^ h

STEP 2

First, we need to calculate the volume of the cone. We can do this by plugging the given values into the formula for the volume of a cone.
Vcone=1πr2hV_{cone} = \frac{1}{} \pi r^2 h

STEP 3

Plug in the given values for the radius and height of the cone to calculate the volume.
Vcone=13π(10m)2(2m)V_{cone} = \frac{1}{3} \pi (10 m)^2 (2 m)

STEP 4

Calculate the volume of the cone.
Vcone=13π(10m)2(2m)=2003πm3V_{cone} = \frac{1}{3} \pi (10 m)^2 (2 m) = \frac{200}{3} \pi m^3

STEP 5

Next, we need to calculate the volume of the cylinder. We can do this by plugging the given values into the formula for the volume of a cylinder.
Vcylinder=πr2hV_{cylinder} = \pi r^2 h

STEP 6

Plug in the given values for the radius and height of the cylinder to calculate the volume.
Vcylinder=π(10m)2(5m)V_{cylinder} = \pi (10 m)^2 (5 m)

STEP 7

Calculate the volume of the cylinder.
Vcylinder=π(10m)2(5m)=500πm3V_{cylinder} = \pi (10 m)^2 (5 m) =500 \pi m^3So, the volume of the cone is 2003πm3\frac{200}{3} \pi m^3 and the volume of the cylinder is 500πm3500 \pi m^3.

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