A cone-shaped paper drinking cup is to be made to hold 24 cm3 of water. Find the height and radius of the cup that will use the smallest amount of paper. (Round your answers to two decimal places.)

I. Radius, r = 2.90 cm

II. Height, h = 4.10 cm

Step-by-step explanation:

Given the following data;

Volume of cone = 24 cm³

To find the height and radius of the cup that will use the smallest amount of paper;

Mathematically, the volume of a cone is given by the formula;

 ......equation 1

Where;

V is the volume of the cone.

r is the radius of the base of the cone.

h is the height of the cone.

Substituting into the formula, we have;

Multiplying both sides by 3, we have;

Making radius, r the subject of formula, we have;

Taking the square root of both sides, we have;

Mathematically, the lateral surface area of a cone is given by the formula;

 ......equation 2

Where;

r is the radius of a cone

l is the slant height of a cone.

To find the slant height, we would apply the Pythagorean' theorem;

Substituting r into the above equation, we have;

Substituting the values of r and l into eqn 2, we have;

Simplifying further, we have;

Next, to find the value of h, we differentiate the above mathematical equation with respect to h;

Limiting w.r.t 0;

Rearranging the equation, we have;

We know that π = 3.142

Cross-multiplying, we have;

Taking the cube root of both sides, we have;

Height, h = 4.10 cm

Lastly, we find the value of r;

Radius, r = 2.90 cm

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