Question

1. the perimeter of a rectangle is 70 cm. if the length is 7 cm longer than the width, what are the dimensions of the rectangle?
2. a medical missions team brought 18 bags of medical supplies with them. the total weight of the bags was 690 pounds. if each carry - on weighed 15 pounds and each checked bag weighed 50 pounds, how many of each type of bag did they bring?

Explanation:

Problem 47

Step1: Define variables

Let the width of the rectangle be \( x \) cm. Then the length is \( x + 7 \) cm.

Step2: Use perimeter formula

The perimeter of a rectangle is given by \( P = 2(l + w) \). Substituting the values, we have \( 70 = 2((x + 7) + x) \).

Step3: Simplify the equation

First, simplify inside the parentheses: \( 70 = 2(2x + 7) \). Then distribute the 2: \( 70 = 4x + 14 \).

Step4: Solve for x

Subtract 14 from both sides: \( 56 = 4x \). Then divide by 4: \( x = 14 \).

Step5: Find length

The length is \( x + 7 = 14 + 7 = 21 \) cm.

Step1: Define variables

Let the number of carry - on bags be \( x \) and the number of checked bags be \( y \).

Step2: Set up equations

We know two things: the total number of bags \( x + y = 18 \) (since there are 18 bags in total) and the total weight \( 15x + 50y = 690 \) (since each carry - on is 15 pounds and each checked is 50 pounds, and total weight is 690 pounds).

Step3: Solve the system of equations

From the first equation, we can express \( x = 18 - y \). Substitute this into the second equation: \( 15(18 - y)+50y = 690 \).

Step4: Simplify the equation

First, distribute the 15: \( 270-15y + 50y = 690 \). Then combine like terms: \( 270 + 35y = 690 \).

Step5: Solve for y

Subtract 270 from both sides: \( 35y = 420 \). Then divide by 35: \( y = 12 \).

Step6: Find x

Substitute \( y = 12 \) into \( x = 18 - y \), so \( x = 18 - 12 = 6 \).

Answer:

The width of the rectangle is 14 cm and the length is 21 cm.

Problem 48