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The diagram shows a right-angled triangular prism ABCDEF - OCR - GCSE Maths - Question 18 - 2019 - Paper 4
Question 18
The diagram shows a right-angled triangular prism ABCDEF. Length AD = 11 cm, length CD = 10 cm and length CF = 6 cm. (a) Calculate the volume of the prism. (b) Us... show full transcript
Worked Solution & Example Answer:The diagram shows a right-angled triangular prism ABCDEF - OCR - GCSE Maths - Question 18 - 2019 - Paper 4
Step 1
(a) Calculate the volume of the prism.
Answer
To calculate the volume of the prism, we use the formula:
- Identify the base area: Since the base is a right triangle with lengths CD and CF,
- Area of triangle = ( \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \times 6 = 30 , \text{cm}^2 )
- Determine the height: The height is given as AD = 11 cm.
- Substitute into the volume formula:
- Volume = ( 30 \times 11 = 330 , \text{cm}^3 )
Step 2
(b) Use trigonometry to show that angle FDC = 31°, correct to the nearest degree.
Answer
To find angle FDC, we can use the tangent function:
- Identify the triangle: In triangle FDC, CD is the opposite side and DF is the adjacent side.
- Use the tangent formula:
- ( \tan(FDC) = \frac{CD}{DF} = \frac{10}{6} )
- Calculate the angle:
- ( FDC = \tan^{-1} \left( \frac{10}{6} \right) \approx 30.9° ), rounding gives 31°. Thus, we conclude that angle FDC = 31°.
Step 3
(c) Calculate the exact length of AF.
Answer
To calculate the length of AF, we apply the Pythagorean theorem:
- Identify lengths:
- CF = 6 cm,
- CD = 10 cm,
- AF is the hypotenuse for triangle AFD.
- Apply the Pythagorean theorem:
- ( AF^2 = AD^2 + DF^2 )
- Substitute the values: ( AF^2 = 11^2 + 10^2 = 121 + 100 = 221 )
- Calculate the exact length:
- ( AF = \sqrt{221} ) which approximates to about 14.83 cm.