BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter T, Problem 4BDT

(a)

To determine

To find: The slope of the line that contains (5,12) and (7,4).

Expert Solution

Answer to Problem 4BDT

The slope of the line is m=43.

Explanation of Solution

Formula used:

The slope of the line is m=y1y2x1x2, where the line passes through the points (x1,y1) and (x2,y2).

Calculation:

It is given that the line that contains (5,12) and (7,4).

Substitute x1=5, y1=12, x2=7, and y2=4 in m=y1y2x1x2 to find the slope of the line,

m=1245(7)=1612=43

Thus, the slope of the line is m=43.

(b)

To determine

To find: The equation and the intercept of the line that contains (5,12) and (7,4).

Expert Solution

Answer to Problem 4BDT

The intercept of the line is 163 and the equation of the line is y=43x163.

Explanation of Solution

Given:

The slope of the line is m=43.

Formula used:

The equation of the straight line is y=mx+c, where m is the slope of the line, c is the x-intercept.

Calculation:

Suppose, the equation of the straight line is y=mx+c.

It is given that, the slope of the line is m=43.

Thus, substitute m=43 in y=mx+c to obtain the equation of the straight line,

y=43x+c

It is given that, the line passes through the point (7,4).

Thus, substitute x=7 and y=4 in y=43x+c to obtain the intercept of the straight line,

4=43(7)+cc=4283c=163

Thus, the value is c=163, that is the intercept of the line.

Thus, substitute c=163 in y=43x+c to obtain the equation of the straight line,

y=43x163

Thus, the intercept of the line is 163 and the equation of the line is y=43x163.

(c)

To determine

To find: The mid points of the points (5,12) and (7,4).

Expert Solution

Answer to Problem 4BDT

The midpoint of the points (5,12) and (7,4) is (1,4).

Explanation of Solution

Formula used:

The midpoint of the points (x1,y1) and (x2,y2) is (x1x22,y1y22).

Calculation:

It is given that the points are (5,12) and (7,4).

Substitute x1=5, y1=12, x2=7, and y2=4 in (x1x22,y1y22) to find the midpoints,

(7+52,4122)=(22,82)=(1,4)

Thus, the midpoint of the points (5,12) and (7,4) is (1,4).

(d)

To determine

To find: The length of the segment between the points (5,12) and (7,4).

Expert Solution

Answer to Problem 4BDT

The length of the segment between the points (5,12) and (7,4) is 20.

Explanation of Solution

Formula used:

The length of the segment between the points (x1,y1) and (x2,y2) is (x1x2)2+(y1y2)2.

Calculation:

It is given that the points are (5,12) and (7,4).

Substitute x1=5, y1=12, x2=7, and y2=4 in (x1x2)2+(y1y2)2 to find the midpoints,

(75)2+(4+12)2=(12)2+(16)2=144+256=400=±20

Length cannot be any negative term. Thus, the length is 20.

Thus, the length of the segment between the points (5,12) and (7,4) is 20.

(e)

To determine

To find: The equation of the perpendicular bisector of AB. (5,12) and (7,4).

Expert Solution

Answer to Problem 4BDT

The equation of the perpendicular bisector is y=34x134.

Explanation of Solution

Given:

The points are A=(5,12) and B=(7,4).

The slope of the line AB is m=43.

The midpoint of the points (5,12) and (7,4) is (1,4).

Formula used:

The equation of the straight line is y=mx+c, where m is the slope of the line, c is the x-intercept.

Calculation:

Suppose, the equation of the straight line is y=mx+c.

It is given that, the slope of the line AB is 43.

Thus, slope of the perpendicular bisector of AB is 34.

Thus, substitute m=34 in y=mx+c to obtain the equation of the straight line,

y=34x+c

It is given that, the line passes through the midpoint, that is (1,4).

Thus, substitute x=1 and y=4 in y=34x+c to obtain the intercept of the straight line,

4=34(1)+cc=4+34c=134

Thus, the value is c=134, that is the intercept of the line.

Thus, substitute c=134 in y=34x+c to obtain the equation of the straight line,

y=34x134

Thus, the equation of the perpendicular bisector is y=34x134.

(f)

To determine

To find: The equation of the circle that has diameter as AB.

Expert Solution

Answer to Problem 4BDT

The equation of the circle is (x+1)2+(y+4)2=100.

Explanation of Solution

Given:

The points are A=(5,12) and B=(7,4).

The midpoint of the points (5,12) and (7,4) is (1,4).

The length of the segment between the points (5,12) and (7,4) is 20.

Formula used:

The equation of the circle is (xh)2+(yk)2=r2, where the center of the circle is at the point (h,k), r is the radius of the circle.

Calculation:

Suppose, the equation of the circle is (xh)2+(yk)2=r2.

It is given that, the center of the circle is at the midpoint of the points (5,12) and (7,4), that is (1,4).

Thus, substitute h=1 and k=4 in (xh)2+(yk)2=r2 to obtain the equation of the straight line,

(x+1)2+(y+4)2=r2

The radius is the half of the length of the segment AB.

It is given that the length of the segment between the points (5,12) and (7,4) is 20.

Thus, the radius is 202=10.

Thus, substitute r=10 in (x+1)2+(y+4)2=r2 to obtain the equation of the straight line,

(x+1)2+(y+4)2=102(x+1)2+(y+4)2=100

Thus, the equation of the circle is (x+1)2+(y+4)2=100.

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