Question

1.3 for use with pages 18-25

1. perimeter the perimeter of the rectangle below is 78 feet. find its dimensions.

image of rectangle with length (2x + 3), width (2x - 4)

1. movie tickets a movie ticket costs $6.50. you have $35.00 to buy tickets and popcorn for four people. how much money is left to buy popcorn after the tickets are paid for?
2. pay rate you need to earn $475 per week to afford the new car you want to purchase. your work week is 45 hours. you get 1.5 times the regular hourly rate for overtime (anything over 40 hours). how much does your hourly rate need to be?
3. car bill the bill for your automobile repairs was $265.74. the cost for labor was $52.00 per hour. the cost for materials was $135.74. how many hours did the mechanic work on your automobile?
4. road trip on friday, you drove 145 miles to stay at your grandmother’s house. on sunday, you returned home and calculated that the round trip travel time was 5 hours. what was your average speed?

Explanation:

Problem 22:

Step1: Recall the perimeter formula for a rectangle.

The perimeter \( P \) of a rectangle is given by \( P = 2\times(\text{length} + \text{width}) \). Here, the length is \( 2x + 3 \) and the width is \( 2x - 4 \), and \( P = 78 \) feet. So we set up the equation:
\[ 2[(2x + 3) + (2x - 4)] = 78 \]

Step2: Simplify the equation inside the brackets.

First, combine like terms inside the brackets: \( (2x + 3) + (2x - 4) = 4x - 1 \). Now the equation becomes:
\[ 2(4x - 1) = 78 \]

Step3: Distribute the 2.

\[ 8x - 2 = 78 \]

Step4: Solve for \( x \).

Add 2 to both sides: \( 8x = 80 \). Then divide both sides by 8: \( x = 10 \).

Step5: Find the dimensions.

Length: \( 2x + 3 = 2(10) + 3 = 23 \) feet.
Width: \( 2x - 4 = 2(10) - 4 = 16 \) feet.

Step1: Calculate the total cost of tickets.

Each ticket costs $6.50, and there are 4 people. So total ticket cost is \( 6.50\times4 \).
\[ 6.50\times4 = 26.00 \]

Step2: Subtract ticket cost from total money.

You have $35.00, so money left for popcorn is \( 35.00 - 26.00 \).
\[ 35.00 - 26.00 = 9.00 \]

Step1: Let the regular hourly rate be \( r \).

For the first 40 hours, you earn \( 40r \). For the overtime (45 - 40 = 5 hours), you earn \( 1.5r \) per hour, so overtime earnings are \( 5\times1.5r = 7.5r \).

Step2: Set up the equation for total weekly earnings.

Total earnings per week should be $475, so:
\[ 40r + 7.5r = 475 \]

Step3: Combine like terms.

\[ 47.5r = 475 \]

Step4: Solve for \( r \).

Divide both sides by 47.5: \( r = \frac{475}{47.5} = 10 \).

Answer:

The length is 23 feet and the width is 16 feet.

Problem 23: