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AQA GCSE · Question 19 · Geometry and Measures
A phone screen is similar to a tablet screen.
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The area of the tablet screen is 420 cm².
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A screen costs £7000 per square metre.
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Work out the cost of a screen for the phone.
A phone screen is similar to a tablet screen.
<br>
The area of the tablet screen is 420 cm².
<br>
A screen costs £7000 per square metre.
<br>
Work out the cost of a screen for the phone.
How to approach this question
1. Find the linear scale factor (LSF) between the tablet and the phone by dividing the corresponding heights.
2. Calculate the area scale factor (ASF) by squaring the LSF.
3. Use the ASF and the area of the tablet to find the area of the phone screen in cm².
4. The cost is given per m². You need to convert this to a cost per cm². Remember 1 m² = 10,000 cm².
5. Multiply the area of the phone screen (in cm²) by the cost per cm² to find the final answer.
Full Answer
**Step 1: Find the linear scale factor (LSF).**
The shapes are similar. The ratio of corresponding lengths gives the LSF.
LSF from tablet to phone = (phone height) / (tablet height) = 6 / 15 = 0.4.
**Step 2: Find the area scale factor (ASF).**
The area scale factor is the square of the linear scale factor.
ASF = (LSF)² = (0.4)² = 0.16.
**Step 3: Calculate the area of the phone screen.**
Area of phone = Area of tablet × ASF
Area of phone = 420 cm² × 0.16 = 67.2 cm².
**Step 4: Convert the cost to £ per cm².**
The cost is given as £7000 per m².
We know 1 m = 100 cm, so 1 m² = 100 cm × 100 cm = 10 000 cm².
Cost per cm² = £7000 / 10 000 cm² = £0.7 per cm².
**Step 5: Calculate the cost of the phone screen.**
Cost = Area of phone × Cost per cm²
Cost = 67.2 cm² × £0.7/cm² = £47.04.
**Answer: £47.04**
**1. Find the Linear Scale Factor (LSF)**
The shapes are similar, so the ratio of their corresponding sides is constant. We can use the heights.
LSF (from tablet to phone) = height of phone / height of tablet = 6 / 15 = 0.4.
**2. Find the Area Scale Factor (ASF)**
The ratio of the areas of similar shapes is the square of the linear scale factor.
ASF = (LSF)² = (0.4)² = 0.16.
**3. Find the Area of the Phone Screen**
Area of phone = Area of tablet × ASF
Area of phone = 420 cm² × 0.16 = 67.2 cm².
**4. Calculate the Cost**
The cost is £7000 per square metre. We need to find the cost for an area of 67.2 cm². We must work in consistent units. Let's convert the cost to be per cm².
1 m = 100 cm
1 m² = (100 cm) × (100 cm) = 10,000 cm².
Cost per cm² = £7000 / 10,000 = £0.70 per cm².
Cost of phone screen = Area of phone × Cost per cm²
Cost = 67.2 × 0.70 = £47.04.
The cost of the screen for the phone is £47.04.
Common mistakes
✗ Using the linear scale factor to calculate the area instead of the area scale factor (e.g., 420 * 0.4).
✗ Incorrectly converting between m² and cm². A common error is to think 1 m² = 100 cm².
✗ Finding the area of the phone correctly but then making a mistake in the cost calculation due to units.
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