Why there is two different answers for the volume of a frustum?

Question

Here is the problem:

A margarine tub has the shape of the frustum of a cone. With the lower base having diameter length 11 cm and the upper base having diameter length 14 cm, the volume of such a container 6 ⅔ cm tall can be determined by using H = 32 ⅔ cm, and h = 26 cm. Find its volume.

Let

R = ¹⁴⁄₂ = 7 r = ¹¹⁄₂ = 5.5 H = 32 ⅔ cm h = 26 cm k = H - h = 6 ⅔

So the solution in the book is using the formula

V = ⅓πR²H - ⅓πr²h

Which is derived from this image:

doing so we have

V = ⅓ π (7 cm)² (32 ⅔ cm) - ⅓ π (5.5 cm)² (26 cm)

V ≈ 852.593 cm³

To check, I used the other formula:

V = ⅓πk (R² + Rr + r²) ; wherein k = H - h

and the result is

V ≈ 822.050 cm³

Why there are different answers, where does the error came from?

Answer 1

The difference comes from the inconsistent information in the question:

• $\dfrac{d}{D}=\dfrac{11}{14}\approx 0.7857$

• $\dfrac{h}{H}=\dfrac{26}{32+\frac23}\approx 0.7723$

If instead you used $h=\frac{11}{3}(6+\frac23)=24+\frac49$ and $H=\frac{14}{3}(6+\frac23)=31+\frac19$ then you would again get $V \approx 822.050$, and consistent answers.