   Chapter 7.1, Problem 80E

Chapter
Section
Textbook Problem

Mechanical Design The surface of a machine part is the region between the graphs of y 1 = | x |   and   y 2 = 0.08 x 2 + k (see figure). (a) Find it such that the parabola is tangent to the graph of y 1 .(b) Find the area of the surface of the machine part.

a)

To determine

To Calculate: The value of ‘k’ if the parabola is tangent to the graph of the curve y1.

Explanation

Given:

The surface of a machine part is the region bounded between the graphs of the curves y1=|x| and y2=0.08x2+k

As shown in figure below:

Formula Used:

Slope of the curve y2 at the point (x0,y0) is 0.16x0

Since functions are even,

Thus the region bounded by curves is symmetric about Y-axis.

So, we consider only first quadrant y1=x.....(1)

If the point (x0,y0) is the point of tangency then from equation (1) y0=x0.

Slope of the tangent line at (x0,y0) is one because it is same at any point of the curve.

Calculation:

Consider the provided figure as given below:

y1=|x| and y2=0.08x2+k

Since the functions are even,

Thus the region bounded by these curves is symmetric about Y-axis.

So, consider only 1st quadrant.

Thus, y1=x ….....(1) x0

b)

To determine

To Calculate: The area of the surface of the machine part.

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