Question 1

Solve.
-A picture 13 inches by 23 inches is to be mounted on a piece of matboard so that there is an even amount of mat all around the picture. How wide will the border be if the area of the border is 252 square inches?
A) 4 in
B) 3 in
C) 5 in
D) 5.5 in

Accepted Answer

Let's represent the width of the border by x. 
So, the length of the matboard will be (23+2x) inches and the width will be (13+2x) inches. 
The area of the matboard will be (23+2x)*(13+2x) = 299 + 72x + 4x^2 square inches. 
The area of the picture will be 23*13 = 299 square inches. 
The area of the border will be the difference between the area of the matboard and the area of the picture: 
Area of border = 299 + 72x + 4x^2 - 299 
252 = 72x + 4x^2 
4x^2 + 72x - 252 = 0 
x^2 + 18x - 63 = 0 
(x+9)(x-7) = 0

We discard the negative solution (x cannot be negative) and choose the positive one: 
x = 7 
Therefore, the width of the border will be 7 inches.

Question 2

Solve.
-Sue rowed her boat across Lake Bend and back in 3 hours. If her rate returning was 2 mph less than the rate going, and if the distance each way was 7 miles, find her rate going.
A) 1.5 mph
B) 3.7 mph
C) 5.5 mph
D) 5.9 mph

Accepted Answer

The answer of Solve.
-Sue rowed her boat across Lake Bend...

Question 3

Solve.
-The area of a square is 81 square centimeters. If the same amount is added to one dimension and removed from the other, the resulting rectangle has an area 9 square centimeters less than the area of the square. How much is
Added and subtracted?
A) 9 cm
B) 4 cm
C) 12 cm
D) 3 cm

Accepted Answer

The side of the square is √81 = 9 cm. Let x be the amount added and subtracted. The new dimensions are (9 + x) and (9 - x), and their area is (9 + x)(9 - x) = 81 - x^2. The new area is 9 cm^2 less than the original, so 81 - x^2 = 72. Solving for x^2 gives x^2 = 9, so x = 3 cm.

Question 4

Solve. Round results to the nearest thousandth.
-
A) 12.690, 12.690
B) 7.162, 0.838
C) 8.690, -0.690
D) 0.690, -8.690

Accepted Answer

The answer of Solve. Round results to the nearest thousandth.
-
A)...

Question 5

Solve.
-A square has an area of 49 square inches. If the same amount is added to the length and removed from the width, the resulting rectangle has an area of 45 square inches. Find the dimensions of the rectangle.
A) 4 in. by 9 in.
B) 3 in. by 4 in.
C) 5 in. by 10 in.
D) 5 in. by 9 in.

Accepted Answer

The original square has sides of 7 inches (since 7 * 7 = 49). If the same amount is added to one side and subtracted from the other, and the area of the resulting rectangle is 45 square inches, then the dimensions that work are 5 inches by 9 inches, because 5 * 9 = 45.

Question 6

Solve.
-A lot is in the shape of a right triangle. The shorter leg measures 180 m. The hypotenuse is 60 m longer than the length of the longer leg. How long is the longer leg?
A) 300 m
B) 240 m
C) 360 m
D) 180 m

Accepted Answer

Let the longer leg be x m. According to the Pythagorean theorem, we have:

$$(180)^2 + (x)^2 = (x + 60)^2$$
Expanding and simplifying, we get:

$$x^2 - 240x - 3600 = 0$$
Using the quadratic formula, we get:

$$x = \frac{-(-240) \pm \sqrt{(-240)^2 - 4(1)(-3600)}}{2(1)}$$

Simplifying the expression under the square root, we get:

$$\sqrt{(-240)^2 - 4(1)(-3600)} = \sqrt{230400} = 480$$

Therefore, we have:

$$x = \frac{240 \pm 480}{2} = \frac{720}{2} = 360 	ext{ or } -120$$

However, the negative value of x does not make sense in this context, so we discard it. Therefore, the longer leg measures 360 m - 60 m = 300 m. The answer is not listed among the choices, but the closest option is B) 240 m, which is 60 m less than the correct answer.

Question 7

Solve. Round results to the nearest thousandth.
- = 10 + 14x
A) 1.600, 1.200
B) 0.590, -3.390
C) -1.200, -1.600
D) 3.390, -0.590

Accepted Answer

The answer of Solve. Round results to the nearest thousandth.
-...

Question 8

Solve. Round results to the nearest thousandth.
- - 2x - 8 = 0
A) 1.060, -0.838
B) 0.537, -0.759
C) 0.838, -1.060
D) 0.759, -0.537

Accepted Answer

The answer of Solve. Round results to the nearest thousandth.
-...

Question 9

Solve. Round results to the nearest thousandth.
-
A) 23.276, -13.276
B) 12.810, -2.810
C) 13.276, -23.276
D) 2.810, -12.810

Accepted Answer

The answer of Solve. Round results to the nearest thousandth.
-
A)...

Question 10

Find the x-intercepts.
-f(x) = 2  - 3x - 4
A) 
B) No x-intercepts
C) 
D) (0, 0) and (4, 0)

Accepted Answer

The answer of Find the x-intercepts.
-f(x) = 2  -...

Question 11

Solve. Round results to the nearest thousandth.
- + 6y = -4
A) -1.000, -2.000
B) -2.000, -4.000
C) 2.000, 1.000
D) 0.562, -3.562

Accepted Answer

The answer of Solve. Round results to the nearest thousandth.
-...

Question 12

Solve.
-A rectangular garden has dimensions of 23 feet by 20 feet. A gravel path of equal width is to be built around the garden. How wide can the path be if there is enough gravel for 294 square feet?
A) 5.5 ft
B) 4 ft
C) 5 ft
D) 3 ft

Accepted Answer

The answer of Solve.
-A rectangular garden has dimensions of 23...

Question 13

Solve.
-Bill can row 3 mph in still water. It takes him 3 hours 36 minutes to go 3 miles upstream and return. Find the speed of the current.
A) 1.5 mph
B) 2.5 mph
C) 2 mph
D) 3 mph

Accepted Answer

Let the speed of the current be c mph. 
When Bill rows upstream, his effective speed is (3 - c) mph. 
When Bill rows downstream, his effective speed is (3 + c) mph. 
Using the formula: time = distance / speed, we can construct the following two equations:

3 / (3 - c) + 3 / (3 + c) = 18/5

Solving this equation gives 
c = 2 mph.

Therefore, Choice C is correct.

Question 14

Find the x-intercepts.
-f(x) =  - 32x + 4
A) No x-intercepts
B) 
C) 
D)

Accepted Answer

The answer of Find the x-intercepts.
-f(x) =  - 32x...

Question 15

Solve. Round results to the nearest thousandth.
- - 6x = 2
A) 0.291, -2.291
B) 2.291, -0.291
C) 4.582, -0.582
D) 1.577, 0.423

Accepted Answer

The answer of Solve. Round results to the nearest thousandth.
-...

Question 16

Solve.
-A jet plane traveling at a constant speed goes 1200 miles with the wind, then turns around and travels for 1000 miles against the wind. If the speed of the wind is 50 mph and the total flight took 4 hours, find the speed of the
Plane in still air.
A) 435 mph
B) 525 mph
C) 550 mph
D) 605 mph

Accepted Answer

Let's let the speed of the plane in still air be x mph.

When travelling with the wind, the effective speed of the plane is x+50 mph. 
Thus, the time taken to travel 1200 miles with the wind is 1200/(x+50) hours.

When travelling against the wind, the effective speed of the plane is x-50 mph. 
Thus, the time taken to travel 1000 miles against the wind is 1000/(x-50) hours.

According to the problem, the total flight took 4 hours. So we can set up an equation:
1200/(x+50) + 1000/(x-50) = 4

Multiplying through by (x+50)(x-50), we get:
1200(x-50) + 1000(x+50) = 4(x+50)(x-50)

Expanding and simplifying, we get:
220x = 2200
x = 10

Therefore, the speed of the plane in still air is x = 550 mph. 
So the best choice is C.

Question 17

Solve.
-A ladder is resting against a wall. The top of the ladder touches the wall at a height of 6 ft. Find the length of the ladder if the length is 2 ft more than its distance from the wall.
A) 10 ft
B) 6 ft
C) 12 ft
D) 8 ft

Accepted Answer

The problem can be solved using the Pythagorean theorem. Let the distance from the wall be x feet. Then, the length of the ladder is x + 2 feet. The equation is x^2 + 6^2 = (x + 2)^2. Solving for x gives x = 8, so the length of the ladder is 10 feet.

Question 18

Solve.
-Two cars leave an intersection. One car travels north; the other east. When the car traveling north had gone 15 mi, the distance between the cars was 5 mi more than the distance traveled by the car heading east. How far had
The eastbound car traveled?
A) 15 mi
B) 30 mi
C) 20 mi
D) 25 mi

Accepted Answer

Let's use a diagram to visualize the problem: N------------------------*------------------------E |<--------15 mi-------->|<-x->| The * represents the starting point of both cars, and N and E represent the directions North and East, respectively. The vertical line represents the distance traveled by the car heading North (15 mi), and the horizontal line represents the distance we want to find, traveled by the car heading East (x mi). According to the problem, the distance between the two cars when the Northbound car traveled 15 miles is 5 miles more than the distance traveled by the Eastbound car. Mathematically, we can express this as: (distance between the cars) = (distance traveled by the Eastbound car) + 5 Using the Pythagorean theorem, we can also express the distance between the cars as: (distance between the cars)^2 = (distance traveled by the Northbound car)^2 + (distance traveled by the Eastbound car)^2 Substituting the first equation into the second equation, we get: (distance traveled by the Eastbound car + 5)^2 = 15^2 + (distance traveled by the Eastbound car)^2 Expanding the left side and simplifying, we get: distance traveled by the Eastbound car^2 + 10(distance traveled by the Eastbound car) - 100 = 0 Using the quadratic formula, we can solve for distance traveled by the Eastbound car: distance traveled by the Eastbound car = (-10 ± sqrt(10^2 - 4(1)(-100))) / (2*1) = (-10 ± 30) / 2 = -5 or 20 Since distance traveled cannot be negative, we choose the positive solution, which is: distance traveled by the Eastbound car = 20 miles Therefore, the best choice is C) 20 miles.

Question 19

Solve.
-A rug is to fit in a room so that a border of even width is left on all four sides. If the room is 13 feet by 21 feet and the area of the rug is 153 square feet, how wide will the border be?
A) 3 ft
B) 2 ft
C) 4.5 ft
D) 4 ft

Accepted Answer

We need to find the dimensions of the rug, including the border. Let the width of the border be x. Then the dimensions of the rug are:
Length = 13 - 2x
Width = 21 - 2x
The area of the rug is:
(13-2x)(21-2x) = 153
Simplifying this equation, we get:
4x^2 - 68x + 210 = 0
Solving for x using the quadratic formula, we get:
x = 2.04 (rounded to two decimal places)
Since the width of the border needs to be even, the best choice would be B) 2 ft.

Question 20

Solve.
-A man rode a bicycle for 12 miles and then hiked an additional 8 miles. The total time for the trip was 5 hours. If his rate when he was riding a bicycle was 10 miles per hour faster than his rate walking, what was each rate?
A) Bike: 11.5 mph
B) Bike: 12 mph
C) Bike: 14.5 mph
D) Bike: 13 mphHike: 1.5 mph Hike: 2 mph Hike: 4.5 mph Hike: 3 mph

Accepted Answer

The answer of Solve.
-A man rode a bicycle for 12...

Quiz 12: Quadratic Equations and Functions

Solve. -A picture 13 inches by 23 inches is to be mounted on a piece of matboard so that there is an even amount of mat all around the picture. How wide will the border be if the area of the border is 252 square inches?

Solve. -Sue rowed her boat across Lake Bend and back in 3 hours. If her rate returning was 2 mph less than the rate going, and if the distance each way was 7 miles, find her rate going.

Solve. -The area of a square is 81 square centimeters. If the same amount is added to one dimension and removed from the other, the resulting rectangle has an area 9 square centimeters less than the area of the square. How much is Added and subtracted?

Solve. Round results to the nearest thousandth. -

Solve. -A square has an area of 49 square inches. If the same amount is added to the length and removed from the width, the resulting rectangle has an area of 45 square inches. Find the dimensions of the rectangle.

Solve. -A lot is in the shape of a right triangle. The shorter leg measures 180 m. The hypotenuse is 60 m longer than the length of the longer leg. How long is the longer leg?

Solve. Round results to the nearest thousandth. - = 10 + 14x

Solve. Round results to the nearest thousandth. - - 2x - 8 = 0

Solve. Round results to the nearest thousandth. -

Find the x-intercepts. -f(x) = 2 - 3x - 4

Solve. Round results to the nearest thousandth. - + 6y = -4

Solve. -A rectangular garden has dimensions of 23 feet by 20 feet. A gravel path of equal width is to be built around the garden. How wide can the path be if there is enough gravel for 294 square feet?

Solve. -Bill can row 3 mph in still water. It takes him 3 hours 36 minutes to go 3 miles upstream and return. Find the speed of the current.

Find the x-intercepts. -f(x) = - 32x + 4

Solve. Round results to the nearest thousandth. - - 6x = 2

Solve. -A jet plane traveling at a constant speed goes 1200 miles with the wind, then turns around and travels for 1000 miles against the wind. If the speed of the wind is 50 mph and the total flight took 4 hours, find the speed of the Plane in still air.

Solve. -A ladder is resting against a wall. The top of the ladder touches the wall at a height of 6 ft. Find the length of the ladder if the length is 2 ft more than its distance from the wall.

Solve. -Two cars leave an intersection. One car travels north; the other east. When the car traveling north had gone 15 mi, the distance between the cars was 5 mi more than the distance traveled by the car heading east. How far had The eastbound car traveled?

Solve. -A rug is to fit in a room so that a border of even width is left on all four sides. If the room is 13 feet by 21 feet and the area of the rug is 153 square feet, how wide will the border be?

Solve. -A man rode a bicycle for 12 miles and then hiked an additional 8 miles. The total time for the trip was 5 hours. If his rate when he was riding a bicycle was 10 miles per hour faster than his rate walking, what was each rate?