Question

translate each problem into an equation. drawing a sketch may help you. 13. 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? 14. 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? 15. the length of a rectangle is twice its width. if the perimeter is 60, find the dimensions of the rectangle. 16. 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. 17. the sides of a triangle have lengths 7, x, and x + 1. if the perimeter is 30, find the value of x. 18. 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:

Step1: Let the shorter piece of ribbon be $x$ feet.

The longer piece is $x + 1$ feet. The sum of the two - pieces is the total length of the ribbon. So the equation is $x+(x + 1)=9$.
Simplify the left - hand side: $2x+1 = 9$.
Subtract 1 from both sides: $2x=9 - 1=8$.
Divide both sides by 2: $x = 4$.
The lengths of the pieces are 4 feet and 5 feet.

Step2: Let the height of the building be $x$ meters.

The height of the tower is $3x$ meters. Since the tower is 50 m taller than the building, the equation is $3x=x + 50$.
Subtract $x$ from both sides: $3x−x=50$, so $2x = 50$.
Divide both sides by 2: $x = 25$. The height of the tower is $3x=75$ meters.

Step3: Let the width of the rectangle be $x$.

The length is $2x$. The perimeter formula for a rectangle is $P = 2(l + w)$. So $2(2x+x)=60$.
Simplify the left - hand side: $2(3x)=6x$. So $6x = 60$.
Divide both sides by 6: $x = 10$. The width is 10 and the length is 20.

Step4: Let the width of the rectangle be $x$.

The length is $x + 1$. The area formula for a rectangle is $A=lw$. So $(x + 1)x=30$, which is $x^{2}+x−30 = 0$.
Factor the quadratic equation: $(x + 6)(x - 5)=0$.
Set each factor equal to zero: $x=-6$ or $x = 5$. Since the width cannot be negative, $x = 5$. The width is 5 and the length is 6.

Step5: The perimeter of a triangle is the sum of its side lengths.

The equation for the triangle's perimeter is $7+x+(x + 1)=30$.
Simplify the left - hand side: $7+x+x + 1=2x+8$. So $2x+8 = 30$.
Subtract 8 from both sides: $2x=30 - 8 = 22$.
Divide both sides by 2: $x = 11$.

Step6: Let the length of each of the equal sides be $x$ cm.

The perimeter of the triangle is $x+x + 15=50$.
Simplify the left - hand side: $2x+15 = 50$.
Subtract 15 from both sides: $2x=50 - 15=35$.
Divide both sides by 2: $x = 17.5$ cm.

Answer:

1. 4 feet and 5 feet
2. 75 meters
3. Width: 10, Length: 20
4. Width: 5, Length: 6
5. $x = 11$
6. 17.5 cm