Answer:
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