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1 Unit 1: Pure Mathematics A

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Question 142

The profile of a specialized parabolic drainage channel is modeled by the curve with equation

h(x)=2x2+256x−100,x>0 h(x) = 2x^2 + \frac{256}{\sqrt{x}} - 100, \quad x > 0 h(x)=2x2+x​256​−100,x>0

where hhh is the depth in millimeters and xxx is the horizontal distance from the left edge of the sensor assembly. The point PPP is the minimum point on the curve.

a.

Use calculus to show that the xxx-coordinate of PPP is 4.

[4]
b.

A horizontal laser beam, represented by the line lll, passes through point PPP and is parallel to the xxx-axis. A shaded region RRR is bounded by the curve, the laser beam lll, and the vertical line x=1x = 1x=1.

Use algebraic integration to find the exact cross-sectional area of the region RRR.

[6]
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1 Unit 1: Pure Mathematics A Questions

  1. A Level
  2. /Maths
  3. /1 Unit 1: Pure Mathematics A

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