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The object below is made from a square-based pyramid joined to a cuboid - OCR - GCSE Maths - Question 8 - 2017 - Paper 1
Question 8
The object below is made from a square-based pyramid joined to a cuboid. The base of the cuboid and the base of the pyramid are both squares of side 4 cm. The heigh... show full transcript
Worked Solution & Example Answer:The object below is made from a square-based pyramid joined to a cuboid - OCR - GCSE Maths - Question 8 - 2017 - Paper 1
Step 1
Calculate the Mass of the Cuboid
Answer
To calculate the mass of the cuboid, we first need to find its volume. The volume ( V ) of the cuboid can be calculated using the formula:
[ V = \text{length} \times \text{width} \times \text{height} ]
Here, both the length and width are 4 cm, and the height is 8 cm:
[ V = 4 \times 4 \times 8 = 128 , cm^3 ]
Next, we use the density of the wood to find the mass of the cuboid:
[ \text{Mass} = \text{Density} \times \text{Volume} ]
The density is 0.67 g/cm³:
[ \text{Mass} = 0.67 \times 128 = 85.76 , g ]
Step 2
Calculate the Volume of the Pyramid
Answer
The total height of the object is given as 13 cm, and the height of the cuboid is 8 cm. Thus, the height of the pyramid can be calculated as:
[ \text{Height of pyramid} = 13 - 8 = 5 , cm ]
The area of the base of the pyramid is:
[ \text{Area} = \text{side}^2 = 4^2 = 16 , cm^2 ]
Now we can find the volume of the pyramid using the volume formula:
[ V = \frac{1}{3} \times \text{Area of base} \times \text{Height} ]
[ V = \frac{1}{3} \times 16 \times 5 = \frac{80}{3} \approx 26.67 , cm^3 ]
Step 3
Calculate the Density of the Granite
Answer
Now we can find the total mass of the object, which is given as 158 g. The mass of the pyramid can be calculated as:
[ \text{Mass of pyramid} = \text{Total mass} - \text{Mass of cuboid} = 158 - 85.76 = 72.24 , g ]
Using the volume of the pyramid calculated earlier (( \approx 26.67 , cm^3 )), we can calculate the density of the granite:
[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} = \frac{72.24}{26.67} \approx 2.71 , g/cm^3 ]