Here is a prism ABCDSPQR. The base ABCD of the prism is a square of side 14 cm. T is the point on BC such that BT : TC = 4 : 3. The cross section of the prism is in the shape of a trapezium of area 147 cm². CR = 12 cm. Find the size of the angle between the line ST and the base ABCD. Give your answer correct to 1 decimal place.

GCSE Edexcel Maths Forming & Solving Equations: Here is a prism ABCDSPQR. The b

Here is a prism ABCDSPQR - Edexcel - GCSE Maths - Question 20 - 2022 - Paper 3

Question 20

Here is a prism ABCDSPQR. The base ABCD of the prism is a square of side 14 cm. T is the point on BC such that BT : TC = 4 : 3. The cross section of the prism is ... show full transcript

Worked Solution & Example Answer:Here is a prism ABCDSPQR - Edexcel - GCSE Maths - Question 20 - 2022 - Paper 3

Step 1

Find TC and BT

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Answer

Given the ratio BT : TC = 4 : 3, let BT = 4x and TC = 3x. The total length BC is given by:

$ext{BC} = 14 ext{ cm}$

Thus,

$4x + 3x = 14 ext{ cm}$

This simplifies to:

$7x = 14 ext{ cm}$

So,

$x = 2 ext{ cm}$

Now, substituting back, we have:

$BT = 4x = 8 ext{ cm}$

$TC = 3x = 6 ext{ cm}$

Step 2

Find CD and area of trapezium

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Answer

To find CD, we use the trapezium area formula:

$ext{Area} = \frac{1}{2} \times (a + b) \times h$

Where:

a = AB = 14 cm
b = CD
h = (ST = CR) = 12 cm

Setting up as follows, we know:

$147 = \frac{1}{2} \times (14 + CD) \times 12$

Solving for CD leads to:

$147 = 6(14 + CD)$

Dividing both sides by 6 gives:

$24.5 = 14 + CD$

Thus:

$CD = 24.5 - 14 = 10.5 ext{ cm}$

Step 3

Find length ST

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Answer

Using Pythagorean theorem in triangle STC:

$ST^2 = (TC)^2 + (SC)^2$

Where:

TC = 6 cm
SC = CD - BT = 10.5 - 8 = 2.5 cm

Thus:

$ST^2 = 6^2 + 2.5^2 = 36 + 6.25 = 42.25$

So,

$ST = \sqrt{42.25} ≈ 6.5 ext{ cm}$

Step 4

Find angle between line ST and base ABCD

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Answer

The angle can be found using:

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{ST}{BT} = \frac{6.5}{8}$

Thus:

$\theta = \tan^{-1}(0.8125) ≈ 39.8^ ext{o}$

Rounding to one decimal place gives:

$\theta ≈ 39.8^ ext{o}$

Here is a prism ABCDSPQR - Edexcel - GCSE Maths - Question 20 - 2022 - Paper 3

Worked Solution & Example Answer:Here is a prism ABCDSPQR - Edexcel - GCSE Maths - Question 20 - 2022 - Paper 3

Find TC and BT

Find CD and area of trapezium

Find length ST

Find angle between line ST and base ABCD

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