6.Bearing-Up is a company that makes steel spheres to be used in bearings.
From the formula list: Volume of sphere \(= \dfrac{4}{3}\pi r^3\); Surface area of sphere \(= 4\pi r^2\)
(a)Bearing-Up melts a steel cylinder of diameter 40 mm and length 1000 mm. It recasts all the steel to make spheres of radius 5 mm.
Diagrams not drawn to scale
Calculate how many spheres can be made from the cylinder. [5]
(b)Bearing-Up also makes spheres with a different radius. Each of these spheres has a surface area of 320\(\pi\) mm2. Calculate the radius of one of these spheres.
Give your answer in the form \(a\sqrt{b}\), where \(a\) is an integer and \(b\) is a prime number. [5]
Mark scheme (a)
Scheme
Marks
Comments
(Volume of cylinder =) π×202×1000 (= 400 000π)
B1
(= 1256000 to 1257000) May be implied in further work
(Volume of sphere =) \(\dfrac{4}{3}\) × π × 53
B1
Or \(\dfrac{500\pi}{3}\). (= 523 to 524)
(Number of spheres made =) \(\dfrac{\pi \times 20^{2} \times 1000}{4/3 \times \pi \times 5^{3}}\) or equivalent
M2
e.g. \(\dfrac{400\,000\pi}{166.6(6)\pi}\) or \(\dfrac{100\,000\pi}{41.6(6)\pi}\) or equivalent M1 for 4/3 × π × 53 × number of spheres = π × 202 × 1000
= 2400 (spheres)
A1
CAO
Mark scheme (b)
Scheme
Marks
Comments
4 × π × r2 = 320π
M1
May be implied in further working
r2 = \(\dfrac{320\pi}{4\pi}\) or (r =) \(\sqrt{\dfrac{320\pi}{4\pi}}\) or equivalent
m1
(r =) \(\sqrt{80}\)
A1
May be implied in further working
= \(4\sqrt{5}\) (mm)
B2
For B2, FT ‘their derived 80’ if possible
For B1, FT 'their derived 80' if possible B1 for writing 80 as a product of 2 or more factors where one of the factors OR the product of 2 of their factors is a square number e.g. 16 × 5, 4 × 20, 2 × 8 × 5, OR B1 for writing \(\sqrt{80}\) as a product of 2 or more factors where one of the factors OR the product of 2 of their factors is a whole number e.g. \(\sqrt{16}\) × \(\sqrt{5}\), \(\sqrt{4}\) × \(\sqrt{20}\), \(\sqrt{2}\) × \(\sqrt{8}\) × \(\sqrt{5}\)
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