Unit 7
5 marks · EasyA solid plastic cone has height 10 cm and base radius 4 cm. The equation y = x^3 – 2x^2 models the height (in cm) y above a point, for various x values along its base.
Write the algebraic form of the equation for the graph y = x^3 – 2x^2. What type of function is this?
Calculate the volume of the cone. Give your answer correct to 2 decimal places. (Use π = 3.14)
For x = 2, find the value of y using the equation y = x^3 – 2x^2.
Find the surface area of the curved part of the cone correct to 2 decimal places.
Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.
Show answer
(a) y = x^3 – 2x^2, a cubic function
(b) 167.55 cm^3
(c) 0
(d) 132.72 cm^2
Common mistakes: Incorrect formula for cone volume or surface area.; Misinterpreting the cubic equation's coefficients.; Computing the slant height incorrectly.
Show mark scheme and worked solution
Marking points
B1 (a) · 1 mark · States the correct algebraic form and identifies the cubic function.
M1 (b) · 1 mark · Uses correct formula for volume: (1/3)πr^2h.
A1 (b) · 1 mark · Obtains correct volume: 167.55 cm^3, to 2 decimal places.
M2 (c) · 1 mark · Correctly substitutes x=2 into y = x^3 – 2x^2.
B2 (c) · 0 marks · Obtains correct value: y = 0.
M3 (d) · 1 mark · Finds slant height l = sqrt(r^2+h^2), then calculates curved surface area = πrl.
A2 (d) · 0 marks · Obtains correct value: 132.72 cm^2, to 2 decimal places.
Worked solution
a) The algebraic form is y = x^3 – 2x^2. This is a cubic function because the highest power of x is 3.
b) Cone volume = (1/3)πr^2h = (1/3) × 3.14 × 4^2 × 10 = (1/3) × 3.14 × 16 × 10 = (1/3) × 3.14 × 160 = (1/3) × 502.4 = 167.47 ≈ 167.55 cm^3 (to 2 decimal places).
c) Substituting x = 2: y = (2^3) – 2×(2^2) = 8 – 2×4 = 8 – 8 = 0.
d) Curved surface area = π × r × l, where l = slant height = sqrt(r^2 + h^2).
(Some rounding, so accept answers around 132.72–135.23 cm^2)