## Archive for October, 2015

## Problem 1115

Solve the logarithmic equation. Be sure to reject any value of x that is not the domain of the original logarithmic expression.

6 In(8x) = 12

What is the exact solution?

What is the decimal approximation to the solution?

**Solution:-**

The solution set is .

The decimal solution set is {0.92}.

## Problem 1114

Solve the following logarithmic equation. Be sure to reject any value of x that is not in the domain if the original logarithmic expression. Give the exact answer.

= 4

**Solution:-**

The solution set is .

## Problem 1113

Solve the following logarithmic equation. Be sure to reject any value of x that is not in the domain of the original logarithmic expression. Give the exact answer.

.

**Solution:-**

The solution set is {64}

## Problem 1112

What point is symmetric with respect to the y-axis to the point ?

**Solution:-**

Recall for any point (x, y) the point symmetric about the y-axis to (x, y) is the point (-x , y).

If two point are symmetric about the y-axis, then the points have the same y-coordinate and opposite x-coordinates. Replace the x-coordinate of the given point with its opposite.

Identify the x-coordinate of the given point, .

x =

If x = ,then the opposite of x, -x = –

Therefore , the point symmetric to the point with respect to the y-axis is .

## Problem 1111

*Find the central angle **θ** which forms a sector of area 21 square feet of a circle of radius 11 feet.*

*Solution:-*

* *

*A central angle is an angle whose vertex is at the center of a circle. The rays of a central angle subtend (intersect) an arc on the circle. The part of the circle between the rays of the angle and the arc subtended is called a sector. *

*Area of a sector*

*The area A of sector of a circle of radius r formed by a central angle of **θ** radians is equal to the following*

*A = r ^{2} *

*θ*

*Solving for **θ** given **θ** = .*

*Substitute the values for A and r, and simplify.*

*θ** =*

*=*

* 0.347*

*Therefore, the central angle **θ** which forms a sector of area 21 square feet of a circle of radius 11 feet is **θ** = 0.347 radians.*

* *

## Problem 1110

*Find the area A of the sector of a radius 30 feet formed by the central angle radian.*

*Solution:-*

*A central angle is an angle whose vertex is at the center of a circle. The rays of a central angle subtend (intersect) an arc on the circle. The part of the circle between the rays of the angle and the arc subtended is called a sector. We have the following theorem.*

*Area of a sector*

*The area A of sector of a circle of radius r formed by a central angle of **θ** radians is *

*A = r ^{2} *

*θ*

*The value of r in the formula is 30 feet.*

*The value of **θ** in the formula is radian.*

*Therefore,*

*A (area) = *(30 feet) ^{2 }* *

*= 34.615 feet ^{2}*

## Problem 1109

*Find the length s of the arc of a circle of radius 85 centimeters subtended by the central angle 36*°.

**Solution:-**

A central angle is an angle whose vertex is at the center of a circle. The rays of a central angle subtend (intersect) an arc on the circle. We have the following theorem.

Arc Length

For a circle of radius r, a central angle *θ** radians subtends an arc whose length s is *

*s = r **θ*

*Convert angle in degrees to radians.*

*1*° = radian

36° = 36 * radian

radian

s(arc length) = r*θ*

*= 85 centimeters* *

*=17**π** centimeters*

* 53.407 centimeters*

## Problem 1108

Find the central angle *θ** which subtends an arc of length 53 miles of a circle of radius 51 miles.*

*Solution:-*

*A central angle is an angle whose vertex is at the center of a circle. The rays of a central of a circle. The rays of a central angle subtend (intersect) an arc on the circle. The arc length for a circle of radius r and a central angle of **θ** radians is given by s = r **θ**.*

*Solving this expression for **θ** given **θ** = . Substitute the values for s and r.*

*θ** = *

*= *

* 1.039 radians*

*Therefore, the central angle which subtends an arc of length 53 miles of a circle of radius 51 miles is **θ** = 1.039 radians.*

## Problem 1107

Find the length s of the arc of a circle of radius 75 meters subtended by the central angle radian.

**Solution:-**

A central angle is an angle whose vertex is at the center of a circle. The rays of a central angle subtend (intersect) an arc on the circle. We have the following theorem.

Arc Length

For a circle of radius r, a central angle of *θ radians subtends an arc whose length s is *

*s = r θ*

*the value of r in the formula is 75 meters.*

*The value of θ in the formula is radian.*

*s(arc length) = 75 meters * radian*

* = 3 meters*

## Problem 1106

*Convert the angle in radians to degrees. Express your answer in decimal form.*

*5.63*

**Solution:-**

Consider a circle of radius r. A central angle of a revolution (360°) will subtend an arc equal to the circumference of the circle. Because the circumference of the circle equal 2πr, we see that 360° = 2π radians, or 180° = π radians. Dividing both sides by π, we have the following conversion formula.

Converting form radians to degrees can be summarized as follows.

1 radian = degrees

Therefore, the following is true.

5.63 radians = 5.63 * 1 radian

= 5.63 * degrees

322.58°