algebra
A gym charges a one-time registration fee plus a monthly membership fee. After 4 months, the total cost is $220. After 10 months, the total cost is $430.
Which equation gives the total cost CC, in dollars, after mm months?
C=35m+80
Explanation:
The function f is defined by
f(x) = (x − 5)² + 9
Which statement is true?
A) The minimum value of f(x) is 5.
B) The minimum value of f(x) is 9.
C) The maximum value of f(x) is 9.
D) The graph has an x-intercept.
B
Explanation:
The function is in vertex form. The vertex is (5, 9), so the minimum value is 9.
A survey of 250 randomly selected students found that 42% preferred online homework.
Based on the survey, about how many of the school's 1,800 students would be expected to prefer online homework?
756
Since the sample is random, use the sample proportion to estimate the entire school.
42% of 1,800
= 0.42 × 1,800
= 756
The circumference of a circle is 24π24π. A sector of the circle has a central angle of 45∘45∘.
What is the length of the arc of the sector?
3π
Explanation:
A 45∘45∘ sector is
45/360 = 1/8
of the entire circle.
Arc length
= (1/8)(24π)
= 3π
Simplify completely.
(5)/(3 + √2)
(15 − 5√2)/7
Explanation:
Multiply by the conjugate.
(3 − √2)/(3 − √2)
Denominator:
9 − 2 = 7
Numerator:
15 − 5√2
Answer:
(15 − 5√2)/7
A system of equations is shown.
3x − 2y = 7
6x + ky = 10
For what value of k does the system have no solution?
-4
Explanation:
For a system to have no solution, the lines must be parallel but have different intercepts.
Rewrite the first equation:
y = (3/2)x − 7/2
Rewrite the second equation:
y = (−6/k)x + 10/k
Set the slopes equal:
−6/k = 3/2
k = -4
If
2(x+3) = 64,
what is the value of x?
3
Explanation:
64 = 2⁶
So
x + 3 = 6
x = 3
The scatterplot for two variables has a correlation coefficient of
r = -0.93.
Which statement is best supported?
A) Increasing x causes y to decrease.
B) There is a strong negative linear association.
C) There is a weak positive association.
D) The variables are unrelated.
B
Explanation:
A correlation coefficient close to -1 indicates a strong negative linear relationship.
Remember:
Triangle ABC is similar to triangle DEF.
The perimeter of triangle ABC is 30 cm, and the perimeter of triangle DEF is 45 cm.
If the area of triangle ABC is 72 cm², what is the area of triangle DEF?
162cm2
Explanation:
The scale factor is
45/30 = 3/2
Areas scale by the square of the scale factor.
72(3/2)²
=72(9/4)
=162
Which expression is equivalent to
√75 − √12 ?
A) √63
B) 3√3
C) √3
D) 3√7
3√3
Explanation:
Simplify each radical.
√75 = 5√3
√12 = 2√3
5√3 − 2√3 = 3√3
The function
f(x) = |3x − 12|
has a minimum value when x equals = ?
4
Explanation:
The absolute value reaches its minimum when the expression inside equals 0.
3x − 12 = 0
x = 4
The polynomial
x³ + kx² − 10x − 24
has a factor of (x − 2).
What is the value of k?
9
Explanation:
Substitute x = 2:
8 + 4k − 20 − 24 = 0
4k = 36
k = 9
A shirt originally costs $64.
Its price is increased by 25% and then decreased by 20%.
What is the final price?
$64
Explanation:
Increase by 25%:
64 × 1.25 = 80
Decrease by 20%:
80 × 0.80 = 64
Notice that a 25% increase followed by a 20% decrease returns to the original price because
1.25 × 0.80 = 1
The diagonals of a rectangle intersect at point P.
One diagonal has length 20 units, and the rectangle has width 12 units.
What is the distance from P to the nearest vertex?
10
Explanation:
The diagonals of a rectangle bisect each other.
Since the diagonal is 20,
the distance from the center to any vertex is
20/2 = 10
The solutions to the equation
x² − 6x + k = 0
are equal.
What is the value of k?
9
Explanation:
Equal solutions occur when the discriminant equals zero.
b² − 4ac = 0
36 − 4k = 0
k = 9
A line passes through the point (-6, 9) and is perpendicular to the graph of
4x + 5y = 30.
The line intersects the x-axis at (a, 0).
What is the value of a?
-66/5
Explanation:
Rewrite the given line:
y = -(4/5)x + 6
The perpendicular slope is 5/4.
Use point-slope form:
y − 9 = (5/4)(x + 6)
Set y = 0.
-9 = (5/4)(x + 6)
-36 = 5(x + 6)
-66 = 5x
x = -66/5
Since the problem asks for the x-coordinate of the intersection, a = -66/5.
The graph of
f(x) = x²
is translated 4 units left and 7 units down.
What equation represents the new graph?
y = (x + 4)² − 7
Explanation:
A translation 4 unites left results in (x-(-4))
A translation 7 units down shifts the entire function - 7
A scientist uses a line of best fit to predict that a plant will be 28 cm tall.
The actual plant measures 33 cm.
What is the residual?
5
Explanation:
Residual = Actual − Predicted
= 33 − 28
= 5
A positive residual means the data point is above the line of best fit.
A right triangle has legs of lengths x and x + 7.
Its hypotenuse has length 17.
What is the value of x?
8
Explanation:
Use the Pythagorean Theorem.
x² + (x + 7)² = 17²
2x² + 14x + 49 = 289
2x² + 14x − 240 = 0
x² + 7x − 120 = 0
(x + 15)(x − 8) = 0
Since side lengths must be positive,
x = 8.
A company purchases two types of storage bins. Small bins cost $18 each, and large bins cost $30 each. The company purchases a total of 40 bins and spends exactly $960.
How many large bins were purchased?
20
Explanation:
Let
s = number of small bins
l = number of large bins
System:
s + l = 40
18s + 30l = 960
Substitute:
18(40 − l) + 30l = 960
720 − 18l + 30l = 960
12l = 240
l = 20
A line passes through (6, 4) and has positive x- and y-intercepts. The x-intercept is three times the y-intercept.
What is the y-intercept?
(0,6)
Since the x-intercept is three times the y-intercept, let
So the line passes through the points
Compute the slope:
m=(0−b)/(3b−0)=−1/3
Now use point-slope form with the given point (6,4):
y−4=−1/3(x−6)
Substitute the y-intercept (0,b):
b−4=−1/3(0−6)
b−4=2
b=6
If f(x) = x² − 8x + 12,
what is the least value of f(x)?
-4
Complete the square:
f(x) = (x − 4)² − 4
Minimum = -4
A company's revenue increased by 12% one year and then decreased by 8% the next year.
Overall, what was the percent change in revenue?
~3%
Explanation:
Percent changes are multiplied, not added.
New revenue:
1.12 × 0.92 = 1.0304
This means the company has
103.04%
of its original revenue.
Overall increase:
103.04% − 100%
= 3.04%, or about 3.0%.
A ladder 26 feet long rests against a vertical wall.
The bottom of the ladder is 10 feet from the wall.
The bottom of the ladder is moved 14 feet farther away from the wall while the top remains against the wall.
By how many feet does the top of the ladder slide downward?
14 ft
Explanation:
Initially:
Height
= √(26² − 10²)
= √576
= 24
After moving:
Distance from wall
= 24
New height
= √(26² − 24²)
= √100
= 10
The ladder moves down
24 − 10
= 14 feet
A closed right circular cylinder has a height that is twice its radius. The cylinder has a volume of 250π cm³.
What is the total surface area of the cylinder, in terms of π?
150π
Explanation:
Since the height is twice the radius, let:
Use the volume formula:
V = πr²h
250π = πr²(2r)
250 = 2r³
125 = r³
r = 5
Since the height is twice the radius,
h = 10
Now use the surface area formula for a closed cylinder:
Surface Area = 2πr² + 2πrh
= 2π(5²) + 2π(5)(10)
= 2π(25) + 100π
= 50π + 100π
= 150π