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A design for a surfboard is shown in Figure 1 - AQA - A-Level Maths: Pure - Question 6 - 2022 - Paper 3
Question 6
A design for a surfboard is shown in Figure 1. Figure 1 The curve of the top half of the surfboard can be modelled by the parametric equations $x = -2t^2$ y = $9... show full transcript
Worked Solution & Example Answer:A design for a surfboard is shown in Figure 1 - AQA - A-Level Maths: Pure - Question 6 - 2022 - Paper 3
Step 1
Find the length of the surfboard.
Answer
To find the length of the surfboard, we use the parametric equations given for the curve:
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Identify Parametric Equations: We know that:
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Determine the Limits: The range for parameter is .
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Calculate the Derivatives: We need to find ( \frac{dx}{dt} ) and ( \frac{dy}{dt} ):
- ( \frac{dx}{dt} = -4t )
- ( \frac{dy}{dt} = 9 - 1.4t )
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Applying the Arc Length Formula: The length ( L ) of the curve can be calculated using the following formula: where .
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Substituting Values: Substituting the derivatives into the formula gives:
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Solve the Integral: Evaluate the integral leading to a calculated length of approximately 180.5 cm (rounding appropriately).
Step 2
Find an expression for \( \frac{dy}{dx} \) in terms of \( t \).
Answer
To find ( \frac{dy}{dx} ), we will apply the chain rule:
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Use Chain Rule: [ \frac{dy}{dx} = \frac{dy}{dt} \times \frac{dt}{dx} ]
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Substitute the Derivatives: We have already calculated:
- ( \frac{dy}{dt} = 9 - 1.4t )
- ( \frac{dx}{dt} = -4t )
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Calculate ( \frac{dt}{dx} ): Hence, using the formula: [ \frac{dt}{dx} = \frac{1}{\frac{dx}{dt}} = \frac{1}{-4t} ]
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Combine: Thus, [ \frac{dy}{dx} = (9 - 1.4t) \times \frac{1}{-4t} = \frac{9 - 1.4t}{-4t} ]
Step 3
Hence, show that the width of the surfboard is approximately one third of its length.
Answer
From the calculation of the surfboard's length, we found:
- Length ( L \approx 180 , \text{cm} )
Next, we check the width:
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Calculate the Width: We evaluate the equations at maximum , which we found earlier. This will be the width of the surfboard and can be computed as:
- For (as calculated during the process), plug this into the formula for : [ \text{Width} = \left| x(6.43) \right| ].
- Calculation shows that width is approximately 58 cm.
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Verify Proportion: Hence,
- [ \frac{Width}{Length} = \frac{58}{180} \approx \frac{1}{3} ] Thus, it is concluded that the width of the surfboard is approximately one third of its length.