Algebra
Easy - linear functions
The function f is defined by f(x) = 80 - 6x. What is the value of f(7)?
38. The value of f(7) can be found by substituting 7 for x in the given function.
f(7) = 80 - 6(7) = 38
f(7) = 38
Easy - Radials and rational exponents: foundations
If 2a = 7th root of 25, what is the value of a?
5/7
Easy - linear and exponential growth
A newborn calf weighs 40 kilograms. Each week its weight increases by 5%. Let W be the weight in kilograms of the calf after t weeks. Which of the following best explains the relationship between t and W?
A. The relationship is linear because W increases by a factor of 5% each time t increases by 1.
B. The relationship is exponential because W increases by a factor of 1.05 each time t increases by 1.
C. The relationship is exponential because W increases by a factor of 5 each time t increases by 1.
D. The relationship is linear because W increases by 2 as t increases from t = 0 to t = 1.
B. The relationship is exponential because W increases by a factor of 1.05 each time t increases by 1.
Easy - area and volume
What is the volume, in cubic meters, of a right circular cone with a radius of 4 meters and a height of 3 meters?
A. 16pi
B. 36pi
C. 40pi
D. 12pi
A. 16pi
Medium - linear inequalities in one or two variables
A truck can haul a maximum weight of 5630 pounds. During one trip, the truck will be used to haul a 190 pound piece of equipment as well as several crates. Some of these crates weigh 25 pounds each and other weigh 62 pounds each. Which inequality represents the possible combinations of the number of 25 pounds crates, x, and the number of 62 pound crates, y, the truck can haul during one trip if only the price of equipment and the crates are being hauled?
A. 25x + 62y < 5440
B. 25x + 62y > 5440
C. 62x + 25y < 5630
D. 62x + 25y > 5630
A.
A truck can haul a maximum of 5,630 pounds, but the truck is also hauling 190 pounds, so it needs to be taken off the total. 5630 - 190 = 5440. A truck can haul at most 5440 pounds (minus equipment), so the < sign would be used.
Medium - radicals and rational exponenets
((a2a-5)/(a-3a0)) x (a2/a3)-4 = ax
What is the value of x?
4
Medium - Center, spread, and shape of distributions
Elena mailed 6 packages, with the following shipping charges.
$6.15, $4.78, $5.61, $4.41, $4.21, $6.55. She mails the 7th package for $5.61, how will that affect the median shipping charge?
A. The median will remain the same.
B. The median will increase by $0.05
C. The median will increase by $0.41
D. The median will increase by $0.60
C. The median will increase by $0.41
Medium - right triangles and trigonometry
A right triangle has legs with lengths of 11 centimeters and 9 centimeters. What is the length of the triangle's hypotenuse, in centimeters?
A. Root 40
B. Root 202
C. 20
D. 202
B. Root 202
Medium - linear equations in two variables
Line r is defined by the equation 4x - 9y = 3. Line s is parallel to line r in the xy-plane. What is the slope of line s?
4/9
convert 4x - 9y = 3 into slope-intercept form y = mx+b because m is the slope.
-9y = -4x + 3
/-9 /-9
y = 4/9x -2/3
M = 4/9, this is your slope of line r, but because parallel lines have equal slopes, the slope of line s is also 4/9.
Medium: linear equations in two variables
Medium - nonlinear functions
The area of a triangle is 70 square centimeters. The length of the base of the triangle is 12 cm greater than the height of the triangle. What is the height, in centimeters of the triangle?
a. 15
b. 18
c. 30
d. 36
B. 18
Medium - ratios, rates, proportional relationships, and units
The population density of Cedar County if 230 people per square mile. The county has a population of 85,100 people. What is the area, in square miles of Cedar County?
370
Medium - area and volume
Square X has a side length of 12 centimeters. The perimeter of square Y is 2 times the perimeter of square X. What is the length, in centimeters, of one side of square Y?
24
Hard - linear functions
x f(x)
1 -64
2 0
3 64
For the linear function f, the table shows three values of x and their corresponding values of f(x). Function f is defined by f(x) = ax + b, where a and b are constraints. What is the value of a - b?
192
In f(x) = ax+ b
a is the slope, found by y2-y1/x2-x1. use two coordinate points from the table: (2, 0) and (3, 64)
(64-0) / (3-2) = 64
B is the y intercept, where the line crosses the y-axis, where x = 0. To find this, substitute a coordinate points from the table into f(x) = ax + b and solve for b
(2, 0)
0 = 64(2) + b
-128 = b
then find a - b
64-(-128) = 192
Hard - linear and quadratic systems
y = x + 4
(y+3)/4 = 7x2 - x + 1
If (a,b) is a solution or the system of equations shown and a > 0 , what is the value of a?
3/7
Hard - one variable data: distributions and measures of center and spread
Data set A consists of the heights of 75 objects and has a mean of 25 meters. Data set B consists of the heights of 50 objects and has a mean of 65 meters. Data set C consists of the heights of the 125 objects from data sets A and B. What is the mean, in meters, of data set C?
41
Hard - Lines, angles, and triangles
In triangles LMN and RST, angles L and R each have a measure of 60 degrees, LN = 10 and RT = 30. Which additional piece of information is sufficient to prove that the triangle LMN is similar to triangle RST?
A. MN = 7 and ST = 7
B. MN = 7 and ST = 21
C. The measures of angles M and S and 70 degrees and 60 degrees, respectively.
D. The measure of angles M and T are 70 degrees and 50 degrees, respectively.
D. The measure of angles M and T are 70 degrees and 50 degrees, respectively.
Hard - Systems of two linear equations in two variables
8x + 7y = 9
24x + 21y = 27
For each real number r, which of the following points lies on the graph of each equation in the xy-plane for the given system?
A. (r, -8r/7 + 9/7)
B. (-8r/7 + 9/7, r)
C. (-8r/7 + 9, 8r/7 + 27)
D. (r/3 + 9, -r/3 + 27)
A.
Divide the second equation by 3 to get 8x + 7y = 9, which is identical to the first equation. The system represents infinitely many solutions along the ling 8x + 7y = 9 and any point (x, y) that satisfies this equation lies on the graph of both equations.
Solve for y in terms of x
8x + 7y = 9
-8x -8x
(7y = -8x + 9 )/7
y = -8/7x + 9/7
If we let x be any real number, r, then the corresponding y value must be y = -8x/7 + 9/7, therefore points are (r, -8x/7 + 9/7), so A is right
Hard - nonlinear equations in one variable and systems of equations in two variables.
y = 2x2 - 21x + 64
y = 3x + a
In the given systems of equations, a is a constant. The graphs of the equations in the given systems intersect at exactly one point, (x, y) in the xy-plane. What is the value of x?
6
Hard - probability and relative frequency: advanced
A customer surveys the stock of gluten-free breads sold in three different grocery stores. At each grocery store, the customer counts the number of different gluten-free breads and compares it to the number of breads that are not gluten-free. The table shows the results of this survey.
Gluten-Free Breads at Different Grocery Stores
Store Gluten-Free Not Gluten-Free
a 2 4
b 7 10
c 3 12
Based on the information in the table, what is the relative frequency of gluten-free breads sold at Store A compared to those sold at all the stores combined?
1/6
Hard - Right triangles and trigonometry
An isosceles right triangle has a perimeter of 94 + 94(root 2) inches. What is the length, in inches, of one leg of this triangle?
A. 47
B. 47(root 2)
C. 94
D. 94(root 2)
B. 47(root 2)