Answer to Question #92553 in Calculus for Tito

Question

What is the smallest perimeter possible for a rectangle whose area is 16 inches?

At what point on the line x=4 does the line segment from (0,0) to(0,6) sibrend the greatest angle?

Answer 1

1)For maxima and minima condition,

The first derivative of perimeter is taken zero

The perimeter of rectangle is 2x+2y

Replace y in the form of x by area of rectangle

xy=16

y=16/x

Thus, perimeter of rectangle(p)=2x+32/x

Taking first derivative of perimeter

P'(x)=2-32x^-2

Taking derivative and setting it equal to zero

0=2-32x^-2

2x²-32=0

x²-16=0

(x-4)(x+4)=0

x-4=0 and x+4=0

Thus,

The value of x found are+-4

Now taking second derivative of perimeter to checking the maxima and minima condition at which value of x having?

P'(x)=2-32x^-2

P"(x)=0+64/x³

P"(x)=64/x³

At x=-4,

P"(x) is negative.

At x=4

P"(x) is positive.

Thus, x=4 is the minima

Thus, x must be +4

Thus

y=16/4=4

Thus, the smallest perimeter is 2x+2y=8+8=16inches

2) the shortest line phase transforming into a member of a level to a line is the perpendicular line phase. point must be on x=4 and other point are on y=6.

For greatest angle, there is the smallest line joining the line x=4 and y=6. The smallest line must be perpendicular to each other. Means from x= 4 to midpoint of y=6 i.e.3.

Thus the pairs of point(4,3)

Another ways of see solution of part B is as follows.

Kindly note that the angle increases as x goes from 0 to √ab and decreases as x increases from √ab. The angle is therefore as large as possible precisely when x = √ab, the geometric mean of a and b.

Where a and b are the points on y axis.

From 0,0 to 0,6, divide a = 3 and b=3 from origin. Thus a=3, b=3.

So for maximum angle subtended is √ab=√(3*3)=√9=3

Thus the point for maximum angle subtended is (4,3).