Question

translate each problem into an equation. drawing a sketch may help you.

1. a ribbon 9 feet long is cut into two pieces. one piece is 1 foot longer than the other. what are the lengths of the pieces?
2. the height of a tower is three - times the height of a certain building. if the tower is 50 m taller than the building, how tall is the tower?
3. the length of a rectangle is twice its width. if the perimeter is 48, find the dimensions of the rectangle.
4. the length of a rectangle is one unit more than its width. if the area is 30 square units, find the dimensions of the rectangle.
5. the sides of a triangle have lengths 7, x, and x + 1. if the perimeter is 30, find the value of x.
6. a triangle has two equal sides and a third side that is 15 cm long. if the perimeter is 50 cm, how long is each of the two equal sides?

Explanation:

13.

Step1: Let the lengths of the pieces

Let the length of one piece be $x$ feet and the other be $x + 1$ feet.

Step2: Set up the equation

The sum of the lengths of the two - pieces is equal to the length of the ribbon. So $x+(x + 1)=9$.

Step3: Simplify the equation

Combining like - terms, we get $2x+1 = 9$.

Step4: Solve for $x$

Subtract 1 from both sides: $2x=9 - 1=8$. Then divide both sides by 2: $x = 4$.

Step5: Find the lengths of the pieces

One piece is $x = 4$ feet and the other is $x + 1=5$ feet.

Step1: Let the height of the building

Let the height of the building be $x$ meters. Then the height of the tower is $3x$ meters.

Step2: Set up the equation

The tower is 50 meters taller than the building, so $3x=x + 50$.

Step3: Solve for $x$

Subtract $x$ from both sides: $3x−x=50$, which simplifies to $2x = 50$. Divide both sides by 2: $x = 25$.

Step4: Find the height of the tower

The height of the tower is $3x$, so $3\times25 = 75$ meters.

Step1: Let the width of the rectangle

Let the width of the rectangle be $w$. Then the length $l = 2w$.

Step2: Recall the perimeter formula

The perimeter of a rectangle $P=2(l + w)$. Given $P = 48$.

Step3: Substitute $l$ into the perimeter formula

$48=2(2w+w)$.

Step4: Simplify the equation

$48=2\times3w=6w$.

Step5: Solve for $w$

Divide both sides by 6: $w = 8$.

Step6: Find the length

$l = 2w=16$.

Answer:

4 feet and 5 feet

14.