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The diagram shows a prism of length 10 cm - OCR - GCSE Maths - Question 20 - 2023 - Paper 2

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The diagram shows a prism of length 10 cm. The cross-section of the prism is a right-angled triangle. The base, b cm, is 2 cm longer than the height, h cm. The volu... show full transcript

Worked Solution & Example Answer:The diagram shows a prism of length 10 cm - OCR - GCSE Maths - Question 20 - 2023 - Paper 2

Step 1

Describe the student's error

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Answer

The student's calculation assumes the height (h) is 4 cm, leading to the conclusion that the base (b) is 6 cm. However, the relationship between b and h is given as b = h + 2. Thus, if h is 4 cm, b should be 6 cm, but the volume calculation does not follow directly from that.

The correct volume formula for a prism is:

V=Base Area×Length V = \text{Base Area} \times \text{Length}

In this case, the area of the triangular cross-section is:

Area=12×b×h \text{Area} = \frac{1}{2} \times b \times h

The volume therefore is:

240=12×b×h×10 240 = \frac{1}{2} \times b \times h \times 10

Step 2

Find the correct value of b

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Answer

From the volume equation, we rewrite it as:

240=5imesbimesh 240 = 5 imes b imes h

This simplifies to:

240=5imes(h+2)×h 240 = 5 imes (h + 2) \times h

Substituting for b (as b = h + 2), we can find a quadratic equation in terms of h:

240=5(h2+2h) 240 = 5(h^2 + 2h)

Expanding gives us:

240=5h2+10h 240 = 5h^2 + 10h

Rearranging, we have:

5h2+10h−240=0 5h^2 + 10h - 240 = 0

Dividing through by 5:

h2+2h−48=0 h^2 + 2h - 48 = 0

Factoring gives:

(h−6)(h+8)=0 (h - 6)(h + 8) = 0

Thus, h = 6 cm (taking the positive solution) and consequently:

b=h+2=6+2=8cm. b = h + 2 = 6 + 2 = 8 cm.

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