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Chapter 12: Problem 1078

Ratio of the area cut off by a parabola \(y^{2}=32 x\) and line \(x=8\) corresponding rectangle contained the area formed by above curves region is....... (a) \((3 / 2)\) (b) \((2 / 3)\) (c) \((1 / 3)\) (d) 3

Short Answer

Expert verified
The ratio of the area cut off by the parabola \(y^2 = 32x\) and the line \(x=8\) to the area of the corresponding rectangle contained in the region formed by these curves is \(\frac{2}{3}\).

Step by step solution

01

Determine the intersection points

We first need to find out the intersection points of the parabola \(y^2 = 32x\) and the line \(x=8\). To do this, we will substitute \(x=8\) into the parabola equation: \(y^2 = 32(8)\) That gives us: \(y^2 = 256\) The intersection points will be: \(y = \pm 16\) So the intersection points are (8, 16) and (8, -16).
02

Calculate the area enclosed by the curves

To find the area enclosed by the parabola and the line, we will use the integral: \(A = \int_{x=a}^{x=b} (y_2 - y_1) dx\) Where \(y_2\) and \(y_1\) are the expressions of the parabola and the line, respectively, in terms of x, and a and b are the limits of integration. We can rewrite the parabola equation as: \(y = \pm \sqrt{32x}\) Now we can find the area enclosed by the curves: \(A = \int_{0}^{8} [(16 - (-16)) - (\sqrt{32x} - (-\sqrt{32x}))] dx\) \(A = \int_{0}^{8} [32 - 2\sqrt{32x}] dx\)
03

Evaluate the integral

Now we need to evaluate the definite integral: \(A = \int_{0}^{8} [32 - 2\sqrt{32x}] dx\) \(A = \left[32x - \frac{4}{3}(32x)^{3/2}\right]_{0}^{8}\) \(A = [256 - \frac{4}{3}(256)^{3/2}] - [0 - \frac{4}{3}(0)^{3/2}]\) \(A = 256 - \frac{1024}{3}\) \(A = \frac{512}{3}\)
04

Calculate the area of the corresponding rectangle

We now need to find the area of the rectangle whose base is length 8 and height is 32, which is: \(R = 8 \cdot 32\) \(R = 256\)
05

Find the ratio of the areas

Now we can find the ratio of the area enclosed by the curves to the area of the corresponding rectangle: \(\text{Ratio} = \frac{A}{R}\) \(\text{Ratio} = \frac{\frac{512}{3}}{256}\) \(\text{Ratio} = \frac{2}{3}\) Therefore, the answer is (b) \(\frac{2}{3}\).

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