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Let f(t)=t^2+4t+2.
Find a value of t such that the average rate of change of f(t) from 0 to t equals 13
.
"average rate of change" is simply the instantaneous slope when t=13.
.
To find it, you can find the derivative of f(t):
f(t)=t^2+4t+2
f'(t)=2t+4
.
Now, find f'(13):
f'(t)=2t+4
f'(13)=2(13)+4
f'(13)=26+4
f'(13)=30

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Starting with:
.

.
.
Multiply by :
.
*
.
.
Doing so yields:

.
.
Now, factoring the denominator:

.
Canceling like-terms we end up with:

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A soccer ball is kicked straight up with an initial velocity of 32 ft. per second. Its height above the earth is given by s (t) = -16t2 + 32t, where,‘s’ is the height in ft. at any time, ‘t’ seconds.
What is the maximum height reached by the ball?
How long after it was thrown up the ball reaches its maximum height?
.
The "vertex" of the parabola will give you both.
.
First, find the "axis of symmetry":
t = -b/2a
t = -32/2(-16)
t = -32/-32
t = 1 second (answer to second question)
.
The "height" is found by plugging the value above back into:
s(t) = -16t^2 + 32t
s(1) = -16(1)^2 + 32(1)
s(1) = -16 + 32
s(1) = 16 feet (answer to first question)

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What is the length of the diagonal of a rectangular bill board with sides of lengths 5 ft and 12 ft.
.
You need to apply Pythagorean theorem:
a^2 + b^2 = c^2
where
a,b are the sides of the right triangle
and
c is the hypotenuse
.
5^2 + 12^2 = c^2
25 + 144 = c^2
169 = c^2
13 ft = c (hypotenuse)

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Scientists have 50g of a radioactive element that decomposes with a half-life of 15 seconds. (3 marks:1 mark each)
a) Write an equation that predicts the amount of the element remaining as a function of time.
P = Ce^(kt)
25 = 50e^(k*15)
25/50 = e^(k*15)
1/2 = e^(k*15)
ln(1/2) = 15k
ln(1/2)/15 = k
Your equation then:
P = 50e^[(ln(1/2)/15)t]
.
b) How much of the element would remain after 2 minutes?
2 minutes = 2*60 = 120 secs
plug it into the equation:
P = 50e^[(ln(1/2)/15)t]
P = 50e^[(ln(1/2)/15)120]
P = 50(.00390625)
P = .195 grams
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c) How long would the element take to decompose to the point where there is only 0.01g remaining?
0.01 = 50e^[(ln(1/2)/15)t]
0.0002 = e^[(ln(1/2)/15)t]
-8.5172 = (ln(1/2)/15)t
-8.5172/(ln(1/2)/15) = t
184.32 secs = t

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Add the two equations together:
-3x+y=7
3x+2y=2
--------
0x+3y= 9
3y = 9
y = 3
.
Substitute the above into equation 1 and solve for x:
-3x+y=7
-3x+3=7
-3x=4
x = -4/3
.
Solution:
x = -4/3
y = 3
Or, (x,y) = (-4/3, 3)

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Ahmad Wishes to purchase mulch to cover his garden. The garden measures 63/8 ft by 81/8 ft. He wants the mulch to be 1/3 ft deep. How much mulch should Ahamad order if he must order a whole number of cubic feet?
My try: I think th ethree frations should be multiplied: (63/8)(81/8)(1/3)= 5103/192
.
You would be correct. However, the part of the problem which states:
"he must order a whole number of cubic feet"
Now, you must convert
5103/192
to a whole number
5103/192 = 26.578125 = 27 cubic feet

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a 2 ft. wire is cut into two pieces. one of the pieces, of length x, is bent into a circle. the other piece is bent into a rectangle whose length is twice the size of its width. what is the total area of the two shapes as a function of x.
.
Let x = circumference of circle
then
2-x = perimeter of rectangle
.
Since for any circle:
circumference = (pi)d = 2(pi)r
then
x = 2(pi)r
r = x/(2(pi))
.
Area of circle:
(pi)r^2
=(pi)[x/(2(pi))]^2]
=(pi)[x^2/(4(pi)^2)]
=x^2/(4(pi))
.
rectangle:
Let w = width
then
2w = length
perimeter = 2(w + 2w)
perimeter = 2(3w)
2-x = 2(3w)
2-x = 6w
(2-x)/6 = w (width)
.
Length:
2w = 2[(2-x)/6] = (2-x)/3 (length)
.
Area of rectangle:
w*L
= (2-x)/6 * (2-x)/3
= (2-x)^2/18
.
Total area:
"area of circle" + "area of rectangle"
= x^2/(4(pi)) + (2-x)^2/18
= 18x^2/(72(pi)) + (4(pi))(2-x)^2/(72(pi))
= [18x^2 + (4(pi))(2-x)^2]/(72(pi))
= [18x^2 + (4(pi))(2-x)(2-x)]/(72(pi))
= [18x^2 + (4(pi))(4-4x+x^2)]/(72(pi))
= [18x^2 + 16(pi) - 16(pi)x + 4(pi)x^2]/(72(pi))
= [18x^2 + 4(pi)x^2 + 16(pi) - 16(pi)x]/(72(pi))
= [2x^2(9 + 2(pi)) + 16(pi)(1-x)]/(72(pi))
= 2[x^2(9 + 2(pi)) + 8(pi)(1-x)]/(72(pi))
= [x^2(9 + 2(pi)) + 8(pi)(1-x)]/(36(pi))
= [x^2(9 + 2(pi)) + 8(pi)(1-x)]/(36(pi))

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P = x^2 + 5x – 50
.
Find out the times when profit is zero:
P = x^2 + 5x – 50
0 = x^2 + 5x – 50
0 = (x+10)(x-5)
x = {-10, 5}
You can throw out the negative answer (you can't sell negative mobile homes) leaving you with:
x = 5
.
Therefore,
x > 5 is your answer

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Area of a room that is:
15 ft by 25 ft
is
15 * 25 = 375 square feet
.
1 square yard = 9 square feet
.
Therefore,
375/9 = 41.67 square yards (or, 41 and 2/3 square yards)

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The results of a survey for an airline are shown below
Traveler Male Female Total
Business 47 72 119
Vacation 71 64 135
Total 118 136 254
Use the chart to find the probability that the traveler was
a) male
"males on business or vacation"/"total travelers"
(47+71)/(119+135)
118/254 * 100
.465 * 100
46.5%
.
b) on vacation given the traveler was female
"female traveling on vacation"/"total female travelers"
64/(72+64) * 100
= 64/136 * 100
= 47.1%
.
c) male given the traveler was on vacation
"males traveling on vacation"/"total travelers on vacation"
71/(71+64) * 100
= 71/135 * 100
= 52.6%

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y = mx + b
is the "slope-intercept" form of a straight line
m is slope
b is the y-intercept at (0,b)
.
Since your equation:
g(x)=-4x-3
is already in "slope-intercept" form -- by inspection we have
y-intercept is at (0,-3)

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Applying "log rules":
logx 16 = -4
16 = x^(-4)
16 = 1/x^4
16x^4 = 1
x^4 = 1/16
x = 1/2

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Renting an Apartment: Three students are planning to rent an apartment for a year and share equally in the cost. By adding a fourth person, each person could save $75 a month. How much is the monthly rent?
.
Let x=rent per student if 3 room mates
then
12(3x) = "total rent for 3 room mates (in a year)"
.
If four students rent
12(4(x-75)) = "total rent for 4 room mates"
.
set them equal and solve for x:
12(3x) = 12(4(x-75))
3x = 4(x-75)
3x = 4x-300
300+3x = 4x
$300 per month = x
.
So, that's $300 per month for each room mate (if there are only 3 of them)
.
Monthly rent = 3(300) = $900 per month

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The problem gives you:
h=-16t^2+64t+80
where
h is height in feet
t is time in secs
.
If it hits the ground, h = 0 so
set h=0 and solve for t:
h=-16t^2+64t+80
0=-16t^2+64t+80
0=-4t^2+16t+5
Solving using the quadratic equation yields:
t = {-0.29, 4.29}
We can throw out the negative solution leaving
t = 4.29 seconds
.
Details of quadratic to follow:

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Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=336 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: -0.29128784747792, 4.29128784747792. Here's your graph:

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Randy has a jar containing 63 coins, which are either quarters or nickels. The total value of the jar is $6.75. How many of each type of coin does he have?
.
Let q = number of quarters
then
63-q = number of nickels
.
.25q + .05(63-q) = 6.75
.25q + 3.15 - .05q = 6.75
.20q + 3.15 = 6.75
.20q = 3.60
q = 3.60/.20
q = 18 (number of quarters)
.
Number of nickels:
63-q = 63-18 = 45

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A 45 foot rope is to be cut into 3 pieces. The second piece must be twice as long as the first piece and the third piece must be 9 feet longer than 3 times the length of the second piece. How long should each of the three pieces be?
.
Good start:
x= the 1st piece
2x= the second piece
From: "the third piece must be 9 feet longer than 3 times the length of the second piece" we have:
3(2x)+9 = the third piece
.
x + 2x + 3(2x)+9 = 45
x + 2x + 6x + 9 = 45
9x + 9 = 45
9x = 36
x = 4 feet (1st piece)
.
2x= 2*4= 8 feet (2nd piece)
.
3(2x)+9 = 3(8)+9 = 24+9 = 33 feet (3rd piece)

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You can't really "solve" it. You can, however, factor it.
.
6x^3+24x
= 6(x^3+4x)
= 6x(x^2+4)

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A card is selected from a deck of 52 playing cards. Find the probability of selecting
A) a seven given the card is not a face card (An ace is not a face card)
Since ALL "seven" cards are NOT face cards, probability is
4/52 = 0.077 = 7.7%
.
B) a spade given the card is red
Since ANY spade is not red:
13/52 = 0.25 = 25%

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A punter can kick a football with an initial velocity of 48 feet per second. How many seconds will it take for the ball to return to the ground?(Hint: Use the formula h=vt-16t2*)
.
No, h=vt-16t^2
is correct
since v (velocity) is feet/sec and if you didn't multiply with time your units would not work.
.
Since h = height (in feet)
set to zero to find out when it hits the ground
.
0 = vt-16t^2
plugging in the given initial velocity:
0 = 48t-16t^2
0 = 3t-t^2
0 = t(3-t)
t = {0, 3}
.
Well, 0 secs is "before" the punter kicked the ball so we're left with:
t = 3 seconds

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Domain specifies the values of 'x' can take.
In this case, you just want to make sure the denominator does NOT equal zero.
.
0 = (x^2+5x+6)
0 = (x+2)(x+3)
x = {-2,-3}
.
Therefore, the domain is
all real numbers except -2 and -3.
.
Or, you can write as:
(-oo, -3) U (-2, -3) U (-3, +oo)
where oo is the infinity symbol

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Strategy:
Put both equation into "point-slope" form to compare slopes:
y = mx + b
where
m is the slope
.
2x-5y=7
-5y=-2x+7
y = (2/5)x - 7/5
and
15y-5=6x
15y=6x+5
y=(6/15)x+5/15
y=(2/5)x+5/15
.
Since both lines have the same slope, the lines are parallel.

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=
.

=
=

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Yes, you are correct.
To check, you could simply plug your answer back into:
log x = 2.5

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3=1/(x+1)^2+2/(x+1)
.
Let t = 1/(x+1)
then, substitute the above in to the original:
3 = t^2 + 2t
0 = t^2 + 2t - 3
0 = (t+3)(t-1)
t = {-3,1}
.
But, remember t = 1/(x+1)
-3 = 1/(x+1)
-3(x+1) = 1
-3x-3 = 1
-3x = 4
x = -4/3
.
1 = 1/(x+1)
(x+1) = 1
x = 0
.
x = {-4/3, 0}

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x^-8-17x^-4+16=0
.
Let t = x^-4
now, substitute the above into the original:
t^2 - 17t + 16 = 0
(t-16)(t-1) = 0
t = {1, 16}
.
To find x, use the fact that t=x^-4
1 = x^-4
1 = 1/x^4
x = 1
.
16 = x^-4
16 = 1/x^4
x^4 = 1/16
x = 1/2
.
x = +-{1, 1/2}

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A boat travels due east for a distance of 15 miles. It then travels due North for a distance of 20 miles, at which point it drops anchor. How many miles is the boat from it's starting point?
.
Applying Pythagorean theorem:
Let d = distance
then
d^2 = 15^2 + 20^2
d^2 = 225 + 400
d^2 = 625
d = sqrt(625)
d = 25 miles

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8x^2y+3y = 2x
Factoring out the y on the left side:
y(8x^2+3) = 2x
Dividing both sides by (8x^2+3):
y = 2x/(8x^2+3)

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4(P-9)=64
Strategy, isolate P:
.
Begin by dividing both sides by 4:
(P-9)=16
P-9 = 16
Add 9 to both sides:
P = 25
.

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f(x)=x^2 + 2x + 5
.
The axis of symmetry is the line x = -b/2a
x = -2/2 = -1
x = -1 (equation of axis of symmetry)
.
f(x)=x^2 + 2x + 5
f(x)=(x^2 + 2x) + 5
f(x)=(x^2 + 2x + __ ) + 5
f(x)=(x^2 + 2x + 1 ) + 5 - 1
f(x)=(x+1)^2 + 4
f(x)=(x-(-1))^2 + 4 (vertex form)
.
vertex is at (-1,4)

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A woman earns 15% more than her husband. Together they make $69,875 per year. What is the husband's annual salary?
.
Let x = husband's annual salary
then
1.15x = wife's annual salary
.
x + 1.15x = 69875
2.15x = 69875
x = 69875/2.15
x = $32,500 (husband's salary)