Question

Case Study - 1

Answer 1

The volume and surface area of the ice cream cone are given by adding

the volume and surface area of the component parts.

Responses (approximate values)

i) 1,696.46 cm³

ii) 229.68 cm²

iii) 169.45 cm³

iv) 565.49 cm²

v) 10 cones

Which methods can be used to calculate the volume and surface area of the given figures?

Given:

The diameter of the cylinder = 12 cm

Height of the cylinder, h = 15 cm

Height of the cone, h = 12 cm

Diameter of the cone, D = 6 cm

Shape of the cap of the cone = Hemispherical

i) The capacity of the container = The volume of the cylinder

Volume of a cylinder = Area of base × height

Radius of the cylinder, R = $\mathbf{\dfrac{D}{2}}$

Which gives;

$R = \dfrac{12 \, cm}{2} = 6 \, cm$

Volume of the cylinder, V = $\pi \times R^2 \times h$

• V = π × (6 cm)² × 15 cm = 540·π cm³ ≈ 1,696.46 cm³

ii) The surface area of a cone, $A_c$ = π·r·(r + √(h² + r²))

Surface area of the hemisphere = 3·π·r²

Which gives;

Surface area of ice cream cone,$A_{ch}$ = π·r·(r + √(h² + r²)) + 3·π·r²

Where;

r = The radius of the cone = $\dfrac{6 \, cm}{2}$ = 3 cm

$A_{ch}$ = π ×3 × (3 + √(12² + 3²)) + 3·π·3² ≈ 229.68

• Surface area of ice cream cone,$A_{ch}$ ≈ 229.68 cm²

iii) The volume of 1 ice-cream, $V_{ic}$ = $\frac{1}{3}$ × π × r² × 12 + $\frac{2}{3}$ × π × r³

Which gives;

$V_{ic}$ = $\frac{1}{3}$ × π × 3² × 12 + $\frac{2}{3}$ × π × 3³≈ 169.45

• The volume of 1 ice-cream, $V_{ic}$ = 169.45 cm³

iv) The CSA of the cylindrical container is given as follows;

Curves Surface Area of a cylinder, CSA = 2·π·R·h

• CSA  = 2 × π × 6 cm × 15 cm ≈ 565.49 cm²

v) The number of ice-cream cones, n, that can be filled is given as follows;

$n = \dfrac{V}{V_{ic}}$

$n = \dfrac{1,696.46 \, cm^3}{169.45 \, cm^3/cone} \approx 10 \ cones$

The number of ice-cream cones that can be filled ≈ 10 cones

Learn more about the volume of regular solids here:

brainly.com/question/13338580