Unit 7
5 marks · EasyTwo solids, A and B, are similar cones. The radius of cone A is cm and the radius of cone B is cm.
The equation of a reciprocal function is given by , where .
Explain how you would use the scale factor between cones A and B to find the ratio of their volumes.
Find the ratio of the volume of cone B to the volume of cone A.
The height of cone B (in cm) is substituted into the reciprocal function above. If the height of cone B is 15 cm, calculate the value of .
Explain what happens to the value of as (the height of cone B) increases. Give a reason using the equation.
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) The ratio of volumes of similar solids is the cube of the ratio of their corresponding lengths (scale factor cubed).
(b) 125 : 8
(c) 1.07
(d) As increases, decreases because in , the denominator gets larger so the value gets smaller.
Common mistakes: Using a linear ratio instead of a cubic for volume scale factor; Incorrect application of reciprocal formula
Show mark scheme and worked solution
Marking points
B1 (a) · 2 marks · States or describes using the cube of the scale factor to find the ratio of volumes, with a clear statement about similarity.
A1 (b) · 1 mark · Correctly finds ratio ; so ratio of volumes is or as a ratio, 125:8.
M1 (c) · 1 mark · Correct substitution of into and valid calculation.
B2 (d) · 1 mark · Correctly reasons that as increases, decreases, with a clear explanation referencing the equation.
Worked solution
a) For similar solids, the ratio of their volumes is equal to the cube of the ratio of any corresponding linear dimensions (for cones, the radius or height).
b) Scale factor = radius of B : radius of A = 10:4 = 2.5:1
This is also 125:8 as an integer ratio.
c) The height of cone B is cm.
d) As increases, decreases, because dividing 16 by a larger number makes the result smaller.