Problem 1 :
A deck has a length 4 feet greater than its width, w. Garrett would like to increase both dimensions of the deck by 2 feet. Which equation represents the new perimeter of the deck ?
A) P(w) = 4 w + 8 B) p(w) = 2w + 4 C) p(w) = 4w + 16
D) p(w) = 8w + 16
Solution :
Width of the rectangle = w
length of the rectangle = w + 4
length and width both increased by 2.
So, new length = w + 4 +2, new width = w + 2
Perimeter = 2(w + 6 + w + 2)
= 2(2w + 8)
= 2(2)(w + 4)
= 4(w + 4)
p(w) = 4w + 16
Problem 2 :
Which equation represents the line with a greater slope and lesser y-intercept then than the line shown ?
A) 3x - y = 4 B) x - 2y = -12 C) 9x + 2y = -10
D) 5x + 3y = 30
Solution :
Option A : 3x - y = 4 y = 3x - 4 Slope = 3, y-intercept = -4 |
Option B : x - 2y = -12 2y = x + 12 y = (1/2)x + 6 Slope = 1/2, y-intercept = 6 |
So, option A is correct.
Problem 3 :
Which expression is equivalent to 49x^{2} - 36 ?
A) (7x - 6) (7x + 6) B) (7x - 6) (7x - 6)
C) (49x + 36) (49x - 36) D) (x - 6) (x + 7)
Solution :
49x^{2} - 36
= (7x)^{2} - 6^{2}
= (7x + 6)(7x - 6)
Problem 4 :
The area of the base of a pyramid and its height are related by the formula V = 1/3 Bh
Which equation finds h, given V by B
A) h = V/3B B) h = 3V/B
C) h = B/3V D) h = V - (B/3)
Solution :
V = 1/3 Bh
Bh = 3V
h = 3V/B
So, option B is correct.
Problem 5 :
The soccer team is selling t-shirts and hats at the foot ball game to earn money.
How much does each t-shirt sell for ?
Solution :
Cost of each t-shirt = x, cost of each hat = y
12x + 5y = 201 -----(1)
6x + 9y = 159 -----(2)
(1) - 2(2)
12x + 5y - 12x - 18y = 201-2(159)
-13y = -117
y = 9
Applying y = 9 in (2), we get
6x + 9(9) = 159
6x = 159 - 81
6x = 78
x = 13
Cost of each t-shirt is $13.
Problem 6 :
Five times Jeff's age plus three times Karen's age equals 92. Karen's age is four more than Jeff's age. How old is Karen ?
Solution :
Let x be Jeff's age. Karen's age = x+4
5x + Karen's age = 92
5x + 3(x + 4) = 92
5x + 3x + 12 = 92
8x = 92 - 12
8x = 80
x = 10
Karen's age = 14
Problem 7 :
Compare the y-intercept of the equation 3x-2y = 14 to the y-intercept in the table. What is the sum of two y-intercepts ?
Solution :
3x - 2y = 14
y-intercept from the equation above,
put x = 0
-2y = 14
y = -7
From the table :
Selecting two points (-6, 2) and (-3, 4)
m = (4 - 2) / (-3 + 6)
m = 2/3
y = (2/3)x + b
The line is passing through the point (-6, 2).
2 = (2/3) (-6) + b
2 = -4 + b
b = 6
Difference between y-intercepts = -7 - 6
= -13
Problem 8 :
What is an x-intercept of the graph of the function
f(x) = x^{2} - 14x + 49 ?
Solution :
f(x) = x^{2} - 14x + 49
x-intercept, put y = 0
x^{2} - 14x + 49 = 0
x^{2} - 7x - 7x + 49 = 0
x(x - 7) -7(x - 7) = 0
(x - 7) (x - 7) = 0
x = 7, x = 7
Problem 9 :
What is the largest three consecutive positive integers if the sum of twice the smallest and the middle integers is equal to twice the largest integer ?
Solution :
Let x, x + 1 and x + 2 are three consecutive positive integers.
2x + x + 1 = 2(x + 2)
3x + 1 = 2x + 4
3x - 2x = 4 - 1
x = 3
Largest number = 3 + 2 ==> 5
Problem 10 :
The function h(t) = -2t^{2} - 7t + 15 models the approximate height of an object at time t. How many seconds will it take the object to hit the ground ?
Solution :
h(t) = -2t^{2} - 7t + 15
When it will hit the ground, h(t) = 0
-2t^{2} - 7t + 15 = 0
-2t^{2} - 10t + 3t + 15 = 0
-2t(t + 5) + 3(t + 5) = 0
(-2t + 3)(t + 5) = 0
t = 3/2, t = -5
It will hit the ground after 1.5 seconds.
Problem 11 :
The function h(t) = -16x^{2} + 64x + 5 represents the height of the object at time x. What is the maximum height of the object reaches ?
Solution :
h(t) = -16x^{2} + 64x + 5
To find the maximum value, we have to express the given quadratic function from standard form to vertex form.
h(t) = -16[x^{2} - 4x] + 5
= -16[x^{2} - 2(x)(2) + 2^{2} - 2^{2}] + 5
= -16[(x - 2)^{2} - 4] + 5
= (x - 2)^{2} + 64 + 5
= (x - 2)^{2} + 69
Vertex is (2, 69). So, the maximum height is 69.
Problem 12 :
The function P(x) = 3500 (0.75)^{x} represents the cost of computer x years after it has been purchased. What percentage of the cost of the computer decreasing each year?
Solution :
P(x) = 3500 (0.75)^{x}
1 - r% = 0.75
r% = 1 - 0.75
r% = 0.25
r% = 25%
25% is decreasing percentage.
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