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Question 1:
At what points (x,y) does the line with equation y = 2x+5 intersect the circle with radius 3 and center
(-2, 0)?
The equation of the circle is (x + 2)2 + y2 = 32
We can then substitute the y for 2x+5
(x - (-2))2 + (2 x + 5)2 - 9 = 0
5 x2 + 24 x + 20 = 0
After using the quadratic formula, to get the x values -1.07335 and -3.72665
We plug that into y=2x+5
y= 2(-1.07335) +5
y= 2.8533
y= 2(-3.72665) +5
y= -2.45
The line intersect the circle at points (-1.07, 2.85) and (-3.73,-2.45)
I don’t know why there is so much space in between the two questions.
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Question 2:
(1) Convert the angle 30° to radians.
I am going to use the degrees to radians formula, which is Degrees *
30
1
*
π
180
=
*
*
180
π
= Degrees
11 π
from radians to degrees
6
I am going to use the radians to degrees formula, which is Radians *
180
π
180
π
= Radians
π
6
(2) Convert the angle
11 π
6
11 π
6
π
180
=
1980
6
= 330°
Question 3:
(1) If cos(θ) =
2
π
and 0 < θ <
find the value of sin(θ).
9
2
In order to solve this problem we have to use the Pythagorean theorem, which is a2 + b2 = c2
22 + b 2 = 9 2
4 + b2 = 81
b=
77
Now that we know the opposite side, we plug in the rest of the numbers.
sin(θ) =
sin(θ) =
77
9
8.77
9
(2) If sin(θ) = -
1
3π
and π < θ <
find the value of cos(θ).
4
2
In order to solve this problem we have to use the Pythagorean theorem, which is a2 + b2 = c2
-12 + b2 = -42
1 + b2 = 16
b=
15
Now that we know the adjacent side, we plug in the rest of the numbers.
cos(θ) = -
15
4
cos(θ) = -
3.87

4
Question 4:
A phone company has a monthly cellular data plan where a customer pays a flat monthly fee and then a
certain amount of money per megabyte (MB) of data used on
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the phone:
• If a customer uses 20 MB, the monthly cost will be $11.20
• If the customer uses 130 MB, the monthly cost will be $17.80
(1) Find a linear equation for the monthly cost of the data plan as a function of the number of MB used.
I am going to make the equation in the form of y=mx+b, monthly cost = (amount of money * MB) +
monthly fee
monthly cost = (0.06 * x) + 10
(2) Interpret the slope and vertical intercept of the equation.
The slope is 0.06 and the vertical intercept is 10.
(3) Use your equation to find the total monthly cost if 250 MB are used.
monthly cost = (0.06 * 250) + 10
monthly cost = $25
Question 5:
A business purchases $125,000 of office furniture which depreciates at a constant rate of 12% each
year.
Find the residual value of the furniture 6 years after purchase.
A = P(1 + r /n)nt
A = 125,000(1 - 0.12)6
A = 58,050.51
Question 6:
A radioactive substance decays exponentially.
A scientist begins with 110 milligrams of the radioactive substance.
After 31 hours, 55 mg of the substance remains.
How many milligrams will remain after 42 hours?
In order to solve this problem, we have to find the half-life and then plug in the time and the initial
amount.
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110 * 2(-42/31)
55
211/31
N
55
211/31

43.0076
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Question 7:
(1) Write a formula for the function g(x) that results when the function f(x) = 2 x2 +1 is horizontally
stretched by a factor of 3, then shifted to the left 4 units and
down 3 units.
g(x) = 6 x2 + 4) - 2
(2) Plot f(x) and g(x) on the same plot.
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f[x_] := 2 x2 + 1
2
g[x_] :=
(x + 4)2 - 2
3
Plot[{f[x], g[x]}, {x, - 10, 10}, PlotStyle → {Red, Green}]
200
150
100
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50
-10
-5
5
10
Question 8:
A scientist begins with 100 mg of a radioactive substance.
After 4 hours, it has decayed to 80 mg.
How long after the process began will it take to decay to 15 mg?
In order to solve this problem, we have to find the half-life using this formula A = P2-t/h 
100 2-4/h  = 80
2-4/h = 80/100
2-4/h = 4/5
log(2-4/h ) = log(4/5)
-4/h =
log(2)
log(4/5)
log(2)
h = -4 log(4/5) 
h = 12.425
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100 * 2(-15/12.425)
43.3095
Question 9:
In the diagram below angles of 30° and -30° are shown, together with cos(30°) - in blue - and sin(-30°) in red:
1.0
0.5
-1.0
0.5
-0.5
1.0
-0.5
-1.0
(1) Given that cos(30°) =
3
and sin(-30°) = - 1 use the addition formula for the sin of a sum of two
2
2
angles, and the fact that 15° = 45°- 30° to calculate sin(15°)
Sin(45 - 30) = Sin(45)Cos(-30) + Cos(45)Sin(-30)
(2) Check you answer with Mathematica®
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Sin[45 Degree] * Cos[- 30 Degree] + Cos[45 Degree] * Sin[- 30 Degree] // N
0.258819
Question 10:
πx
1
) and y =
on the same plot, and so determine for how many values of x between
4
2
πx
1
-2π and 2π we have sin(
)=
4
2
(1) Plot y = sin(
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PlotSin[(π * x) / 4], 1  2, {x, - 2 π, 2 π}
1.0
0.5
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-6
-4
2
-2
4
6
-0.5
-1.0
(2) Find the exact value of x between 0 and 2 for which sin(
πx
1
)=
4
2
x = 2/3
Question 11:
The more you study for a certain exam, the better your performance on it:
• If you study for 10 hours, your score will be 65%.
• If you study for 20 hours, your score will be 95%.
You can get as close as you want to a perfect score just by studying long enough.
Assume your percentage score, p(n), is a function of the number of hours, n, that you study in the form
p(n) =
n+b
n+d
(1) If you want a score of 80%, how long do you need to study?
If 20 = 95 and 10 = 65, then 15 must = 80
15 hours
(2) Plot a graph showing the behavior of p(n) as a function of n for 0 ≤ n ≤ 40.
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Plot- 8.916 + n  - 8.333 + n, {n, 0, 40}
1.3
1.2
1.1
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0.9
0.8
0.7
10
20
30
40
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Question 12:
The number of people P(t) in a town who have heard a rumor after t days can be modeled by the
function P(t) =
500
1+49 ⅇ-0.7 t
(1) Plot a graph of this equation.
Plot500  1 + 49  E ^ 0.7 * t, {t, - , 4}
120
100
80
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60
40
20
-3
-2
-1
1
2
3
4
(2) How many people started the rumor?
10 people
(3) How many people have heard the rumor after 3 days?
70 people
(4) How long will it take until 300 people have heard the rumor?
6.2 days
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