Higher June 2022 Paper 2 Q17
17 A solid cone is joined to a solid hemisphere to make the solid T shown below.
| Volume of sphere = \(\dfrac{4}{3}\pi r^3\) Volume of cone = \(\dfrac{1}{3}\pi r^2 h\) |  |
The diameter of the base of the cone is 7 cm.
The diameter of the hemisphere is 7 cm.
The total volume of T is \(120\pi\) cm3
The total height of T is \(y\) cm.
Give your answer correct to 3 significant figures. (4)
The diameter of the base of the cone and the diameter of the hemisphere are both increased by the same amount.
Assuming the total volume of T does not change,
| Answer | Mark | Mark scheme |
|---|---|---|
| 25.9 | P1 | for process to find volume of hemisphere, eg \(\dfrac{1}{2} \times \dfrac{4}{3} \times \pi \times 3.5^3\ (= 89.797\ldots)\ \left(\dfrac{343\pi}{12}\right)\) or for a correct expression for the volume of the cone, eg \(\dfrac{1}{3} \times \pi \times 3.5^2(y - 3.5)\) or \(\dfrac{1}{3} \times \pi \times 3.5^2 \times h\) |
| P1 | for setting up an equation linking all three aspects, eg \(\dfrac{1}{2} \times \dfrac{4}{3} \times \pi \times 3.5^3 + \dfrac{1}{3} \times \pi \times 3.5^2(y - 3.5) = 120\pi\) or \(\text{``}89.797\ldots\text{''} + \text{``}12.828\ldots\text{''}(y - 3.5) = \text{``}376.99\ldots\text{''}\) or \(\text{``}28.5833\ldots\text{''}\pi + \text{``}4.0833...\text{''}\pi(y - 3.5) = 120\pi\) | |
| P1 | for process to isolate \(y\) or \((y - 3.5)\) or \(h\) in their equation, eg \(\dfrac{120\pi - \dfrac{1}{2} \times \dfrac{4}{3}\pi 3.5^3 + \dfrac{1}{3}\pi 3.5^3}{\dfrac{1}{3}\pi 3.5^2}\) or \(\dfrac{\text{``}376.99...\text{''} - \text{``}89.797...\text{''} + \text{``}44.898...\text{''}}{\text{``}12.828...\text{''}}\) or \(\dfrac{120\pi - \text{``}28.583...\text{''}\pi + \text{``}14.291...\text{''}\pi}{\text{``}4.083...\text{''}\pi}\) oe | |
| A1 | for answer in range 25.8 to 26.3 SCB3 for an answer in the range 22.3 to 22.8 or \(\dfrac{1097}{49}\) |
Additional guidance
‘\(y - 3.5\)’ may be seen as a new variable, but cannot be just \(y\)
Condone missing brackets
Accept decimals rounded or truncated to 1dp
\(120\pi - \text{``}89.797...\text{''} = 287.193...\) or \(\dfrac{1097\pi}{12}\)
\(\pi\) may be missing throughout
Award of this mark implies award of the previous
May be seen in multiple steps
If an answer is given in the range in working and then rounded incorrectly award full marks.
| Answer | Mark | Mark scheme |
|---|---|---|
| explanation | C1 | for explanation, eg Acceptable examples the height would decrease the height would be 0 at 14.227 \(y\) would be smaller it would decrease Not acceptable examples the height would increase |