Question 1

Solve the problem.
-A parallelogram has a base of length 3x + 1 and a height of x + 1 and has an area of 96 square units. Find the base and height of the parallelogram. (A = bh)
A) height = 2 units; base = 48 units
B) height = 4 units; base = 24 units
C) height = 8 units; base = 12 units
D) height = 6 units; base = 16 units

Accepted Answer

The answer of Solve the problem.
-A parallelogram has a base...

Question 2

Solve the problem.
-A triangle has a base of length x + 2 and a height of x + 8 and has an area of 36 square units. Find the base and height. (A =  bh)
A) base = 6 units; height = 12 units
B) base = 6 units; height = 6 units
C) base = 4 units; height = 9 units
D) base = 8 units; height = 9 units

Accepted Answer

The answer of Solve the problem.
-A triangle has a base...

Question 3

Solve the problem.
-A triangle has a base of length 3x + 1 and a height of x + 6 and has an area of 45 square units. Find thebase and height. (A =  bh)
A) base = 3 units; height = 15 units
B) base = 9 units; height = 5 units
C) base = 10 units; height = 9 units
D) base = 3 units; height = 30 units

Accepted Answer

The answer of Solve the problem.
-A triangle has a base...

Question 4

Solve the problem.
-A triangle has a base of length 2x + 3 and a height of x + 3 and has an area of 27 square units. Find thebase and height. (A =  bh)
A) base = 9 units; height = 3 units
B) base = 9 units; height = 6 units
C) base = 3 units; height = 9 unit
D) base = 18 units; height = 3 units

Accepted Answer

The answer of Solve the problem.
-A triangle has a base...

Question 5

Solve the problem.
-The length of a rectangle is 6 inches more than its width. If 3 inches are taken from the length and added to the width, the figure becomes a square with an area of 169 square inches. What are the dimensions of the original figure?
A) 7 in. by 13 in.
B) 10 in. by 16 in.
C) 10 in. by 13 in.
D) 13 in. by 13 in.

Accepted Answer

The side of the square is the square root of 169, which is 13 inches. Originally, the length was 3 inches more than 13, making it 16 inches, and the width was 3 inches less than 13, making it 10 inches.

Question 6

Solve the problem.
-The length of a rectangular frame is 5 cm more than the width. The area inside the frame is 126 square cm. Find the width of the frame.
A) 19 cm
B) 11 cm
C) 14 cm
D) 9 cm

Accepted Answer

The width of the frame can be found by setting up the equation w(w + 5) = 126, where w is the width and w + 5 is the length. Solving this quadratic equation, we find that w = 9 cm is the correct width that satisfies the given conditions.

Question 7

Solve the problem.
-The height of a box is 5 inches. The length is three inches more than the width. Find the width if the volume is 440 cubic inches.
A) 5 in.
B) 88 in.
C) 8 in.
D) 11 in.

Accepted Answer

The volume of a box is given by the formula length × width × height. Let the width be x inches. Then, the length is x + 3 inches. Given the height is 5 inches and the volume is 440 cubic inches, we have the equation: $x(x + 3)(5) = 440$. Simplifying, we get $5x^2 + 15x - 440 = 0$. Dividing the entire equation by 5 gives $x^2 + 3x - 88 = 0$. Factoring this equation, we find $x = 8$ or $x = -11$. Since a width cannot be negative, the width is 8 inches.

Question 8

Solve the problem.
-The height of a box is 4 inches. Its length is 4 inches more than its width. Find the length if the volume is 180 cubic inches.
A) 9 in.
B) 45 in.
C) 5 in.
D) 4 in.

Accepted Answer

The volume of a box is given by the formula length × width × height. Let the width be x inches. Then, the length is x + 4 inches. Given the height is 4 inches and the volume is 180 cubic inches, we can set up the equation: $x(x + 4)(4) = 180$. Simplifying gives $4x^2 + 16x - 180 = 0$, which simplifies further to $x^2 + 4x - 45 = 0$. Factoring this equation gives $(x + 9)(x - 5) = 0$, so $x = 5$ inches (since width cannot be negative). Therefore, the length is $x + 4 = 5 + 4 = 9$ inches.

Question 9

Solve the problem.
-Doris has a rectangular fish pond 9 by 12 feet. She wants to put grass in a strip of uniform width around the pond. She has enough grass seed for 162 square feet. How wide will the strip be?
A) 3 ft
B) 5 ft
C) 4 ft
D) 5.5 ft

Accepted Answer

The area of the pond is 9 * 12 = 108 square feet. The total area, including the grass, is 108 + 162 = 270 square feet. Let the width of the grass strip be x feet. The dimensions of the new area are (9+2x) by (12+2x). So, (9+2x)(12+2x) = 270. Solving for x gives x = 3 feet.

Question 10

Solve the problem.
-Suppose a pyramid has a rectangular base whose width is 3 centimeters less than its length. If the height is 9 centimeters and the volume is 162 cubic centimeters, find the length of the base. 
A) 54 cm
B) 3 cm
C) 6 cm
D) 9 cm

Accepted Answer

The answer of Solve the problem.
-Suppose a pyramid has a...

Question 11

Solve the problem.
-A square piece of cardboard is to be formed into an open-topped box by cutting  squares from the corners and folding up the sides. The volume of the resulting box is given by the formula  , where x is the original length of each side of the piece of cardboard. What is the volume of the box if the original length of each side of the piece of cardboard is 16 inches?
A) 100 
B) 768 
C) 30 
D) 300

Accepted Answer

The answer of Solve the problem.
-A square piece of cardboard...

Question 12

Solve the problem.
-The product of two consecutive integers is 11 more than their sum. Find the integers.
A) 3, 4 or -3, -2
B) 4, 5
C) -3, -2
D) 4, 5 or -3, -2

Accepted Answer

The integers can be found by setting up the equation: n(n+1) = n + (n+1) + 11. Solving this gives n^2 + n = 2n + 12, which simplifies to n^2 - n - 12 = 0. Factoring gives (n-4)(n+3) = 0, so n = 4 or n = -3. The consecutive integers are 4, 5 or -3, -2.

Question 13

Solve the problem.
-Find three consecutive integers such that the sum of the squares of the smaller two is equal to the square of the largest.
A) -1, 0, 1
B) 3, 4, 5
C) -1, 0, 1 or 3, 4, 5
D) -3, -4, -5

Accepted Answer

The problem can be solved algebraically by letting the three consecutive integers be $x$, $x+1$, and $x+2$. The equation based on the given condition is $x^2 + (x+1)^2 = (x+2)^2$. Solving this equation, we find that it holds true for $x = -1$ and $x = 3$, which correspond to the sets of integers $-1, 0, 1$ and $3, 4, 5$ respectively. Therefore, both sets of integers satisfy the condition.

Question 14

Solve the problem.
-Find three consecutive odd integers such that the sum of all three is 36 less than the product of the smaller two.
A) 7, 9, 11
B) 7, 9, 11, or -6, -4, -2
C) 5, 7, 9
D) -6, -4, -2

Accepted Answer

Let the three consecutive odd integers be $x$, $x+2$, and $x+4$. The sum of these integers is $x + (x+2) + (x+4) = 3x + 6$. The product of the smaller two is $x(x+2)$. According to the problem, $3x + 6 = x(x+2) - 36$. Simplifying, we get $3x + 6 = x^2 + 2x - 36$, which simplifies further to $x^2 - x - 42 = 0$. Factoring, we find $(x-7)(x+6) = 0$, so $x = 7$ or $x = -6$. Thus, the sets of integers are $7, 9, 11$ or $-6, -4, -2$. However, since the question asks for odd integers, the correct answer is $7, 9, 11$.

Question 15

Solve the problem.
-Find three consecutive integers such that the square of the sum of the smaller two is 72 more than the square of the largest.
A) -5, -4, -3
B) 5, 6, 7, or -5, -4, -3
C) 5, 6, 7
D) 3, 5, 7

Accepted Answer

Let the three consecutive integers be $x$, $x+1$, and $x+2$. The problem states that $(x + (x+1))^2 = (x+2)^2 + 72$. Simplifying gives $4x^2 + 4x + 1 = x^2 + 4x + 4 + 72$, which simplifies to $3x^2 - 75 = 0$. Solving this quadratic equation gives $x = 5$ or $x = -5$, leading to the sets of integers $5, 6, 7$ and $-5, -4, -3$.

Question 16

Solve the problem.
-The product of two consecutive integers is 8 less than 8 times their sum. Find the integers.
A) 0, 1
B) 0, 1 or 16, 17
C) 15, 16
D) 0, 1 or 15, 16

Accepted Answer

Let the two consecutive integers be $x$ and $x+1$. The equation based on the given information is $x(x + 1) = 8(x + x + 1) - 8$. Simplifying gives $x^2 + x = 16x + 8 - 8$, which simplifies further to $x^2 - 15x = 0$. Factoring out $x$, we get $x(x - 15) = 0$, which gives the solutions $x = 0$ and $x = 15$. Therefore, the pairs of integers are $0, 1$ and $15, 16$.

Question 17

Solve the problem.
-A long-distance runner runs 4 miles south and then 9 miles east. How far is the runner from the starting point? Round approximations to the nearest tenth.
 
East --9 mi-->
A) 4.9 mi
B) 48.5 mi
C) 6.5 mi
D) 9.8 mi

Accepted Answer

The answer of Solve the problem.
-A long-distance runner runs 4...

Question 18

Solve the problem.
-The diagram below shows the side view of a plan for a slanted roof. Find the unknown length in this roof plan. Round approximations to the nearest tenth.
 
20 ft
A) 312.5 ft
B) 17.5 ft
C) 12.5 ft
D) 25 ft

Accepted Answer

The answer of Solve the problem.
-The diagram below shows the...

Question 19

Solve the problem.
-A boat travels 7 miles south and then 9 miles east. How far is the boat from its starting point? Round approximations to the nearest tenth.
 
East --9 mi-->
A) 5.7 mi
B) 8 mi
C) 11.4 mi
D) 65 mi

Accepted Answer

The answer of Solve the problem.
-A boat travels 7 miles...

Question 20

Solve the problem.
-The diagram below shows a rope connecting the top of a pole to the ground. The rope is 24 yd long and touches the ground 17 yd from the pole. How tall is the pole? Round approximations to the nearest tenth.
  
17 yd
A) 143.5 yd
B) 8.5 yd
C) 20.5 yd
D) 16.9 yd

Accepted Answer

The answer of Solve the problem.
-The diagram below shows a...

Quiz 7: Rational Expressions and Applications

Solve the problem. -A parallelogram has a base of length 3x + 1 and a height of x + 1 and has an area of 96 square units. Find the base and height of the parallelogram. (A = bh)

Solve the problem. -A triangle has a base of length x + 2 and a height of x + 8 and has an area of 36 square units. Find the base and height. (A = bh)

Solve the problem. -A triangle has a base of length 3x + 1 and a height of x + 6 and has an area of 45 square units. Find thebase and height. (A = bh)

Solve the problem. -A triangle has a base of length 2x + 3 and a height of x + 3 and has an area of 27 square units. Find thebase and height. (A = bh)

Solve the problem. -The length of a rectangle is 6 inches more than its width. If 3 inches are taken from the length and added to the width, the figure becomes a square with an area of 169 square inches. What are the dimensions of the original figure?

Solve the problem. -The length of a rectangular frame is 5 cm more than the width. The area inside the frame is 126 square cm. Find the width of the frame.

Solve the problem. -The height of a box is 5 inches. The length is three inches more than the width. Find the width if the volume is 440 cubic inches.

Solve the problem. -The height of a box is 4 inches. Its length is 4 inches more than its width. Find the length if the volume is 180 cubic inches.

Solve the problem. -Doris has a rectangular fish pond 9 by 12 feet. She wants to put grass in a strip of uniform width around the pond. She has enough grass seed for 162 square feet. How wide will the strip be?

Solve the problem. -Suppose a pyramid has a rectangular base whose width is 3 centimeters less than its length. If the height is 9 centimeters and the volume is 162 cubic centimeters, find the length of the base.

Solve the problem. -The product of two consecutive integers is 11 more than their sum. Find the integers.

Solve the problem. -Find three consecutive integers such that the sum of the squares of the smaller two is equal to the square of the largest.

Solve the problem. -Find three consecutive odd integers such that the sum of all three is 36 less than the product of the smaller two.

Solve the problem. -Find three consecutive integers such that the square of the sum of the smaller two is 72 more than the square of the largest.

Solve the problem. -The product of two consecutive integers is 8 less than 8 times their sum. Find the integers.

Solve the problem. -A long-distance runner runs 4 miles south and then 9 miles east. How far is the runner from the starting point? Round approximations to the nearest tenth. East --9 mi-->

Solve the problem. -The diagram below shows the side view of a plan for a slanted roof. Find the unknown length in this roof plan. Round approximations to the nearest tenth. 20 ft

Solve the problem. -A boat travels 7 miles south and then 9 miles east. How far is the boat from its starting point? Round approximations to the nearest tenth. East --9 mi-->

Solve the problem. -The diagram below shows a rope connecting the top of a pole to the ground. The rope is 24 yd long and touches the ground 17 yd from the pole. How tall is the pole? Round approximations to the nearest tenth. 17 yd