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The mass m(t) remaining after t hours from a 50 gram sample of a radioactive substance is given by the equation m(t)= 50e^-0.0578t. after how long will 30 grams of substance remain?
.
m(t)= 50e^-0.0578t
the problem gives m(t) as 30 you simply need to solve for t:
30= 50e^-0.0578t
30/50 = e^-0.0578t
ln(30/50) = -0.0578t
(ln(30/50))/-0.0578 = t
8.838 hours = t

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Batman and Robin working together can capture the Joker in 3 hours. It takes Robin, working alone, 1 more hour than Batman to capture the Joker. Find how long it takes Robin to capture the Joker working alone and how ling it takes Batman to capture the joker working alone.
.
Let x = hours it takes Batman to work alone
then
x+1 = hours it takes Robin to work alone
.
3(1/x + 1/(x+1)) = 1
3(x+1 + x) = x(x+1)
3(2x+1) = x(x+1)
6x+3 = x^2+x
0 = x^2+x-6x-3
0 = x^2-5x-3
Solving using the quadratic equation we get:
x = {5.5414, -0.5414}
We can toss out the neg answer leaving:
x = 5.5414 hours (time it takes Batman working alone)
.
Robin:
x+1 = 5.5414+1 = 6.5414 hours
.
Details of quadratic:

Solved by pluggable solver: SOLVE quadratic equation with variable

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=37 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 5.54138126514911, -0.54138126514911. Here's your graph:

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I assume you meant:
5e^(.2x)=7
e^(.2x) = 7/5
.2x = ln(7/5)
x = ln(7/5)/.2
x = 1.682

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cos-1 (√2\2)
is asking you what angle x would be if
cos(x) = √2\2
.
In this case, looking at the "unit circle" it would be
(pi/4) radians
or
45 degrees

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Factoring:

x = {-1/2, 2}

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simply divide 250 by 60
.
250/60 = 4.167 hours

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x^2 - 64
can be rewritten as:
x^2 - 8^2
this is a "special case" -- "difference of squares"
which factors out to
(x-8)(x+8)

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A salesman sold $300 more than the rest of the sales staff combined. If the sales total for the day was $2500, then how much did the rest of the sales staff sell?
.
Let x = amount "rest of sales staff sold"
then
x+300 = amount "salesman sold"
.
"rest of sales staff sold" + "salesman sold" = total sales
x + (x+300) = 2500
2x + 300 = 2500
2x = 2200
x = $1100

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log(base)3 (x+6) + log(base)3 (x-6) - log(base)3 x =2
log(base)3 (x+6)(x-6) - log(base)3 x =2
log(base)3 [(x+6)(x-6)]/x =2
[(x+6)(x-6)]/x = 3^2
[(x+6)(x-6)]/x = 9
(x+6)(x-6) = 9x
x^2 - 36 = 9x
x^2 - 9x - 36 = 0
(x-12)(x+3) = 0
x = {-3, 12}
.
We can throw out the -3 -- it's an extraneous solution -- leaving us with:
x = 12

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e^2x + e^x -6=0
.
This is simply a "quadratic equation":
.
Let y = e^x
then
y^2 = (e^x)^2 = (e^x)(e^x) = e^2x
.
So now we can rewrite:
e^2x + e^x -6=0
y^2 + y -6=0
(y+3)(y-2) = 0
y = {-3, 2}
.
Since we said:
y = e^x
ln y = x
.
So,
x = ln 2 (this is your solution)

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Quadrant II
.
Tan - negative in II and IV
Sin - positive in I and II

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1.strategy to solve equations linear
In each, simply isolate the unknown to one side of the equation -- doesn't matter which side. To check your answer, plug it back into the original equation.
v-4(-v)=-2(2v-1)
distribute the 4 and -2 to each term inside parentheses:
v+4v=-4v+2
combine the v's on the left:
5v=-4v+2
add 4v to both sides:
9v = 2
finally, divide both sides by 9:
v = 2/9
9-2m/7=19
subtract 9 from both sides:
-2m/7=10
multiply both sides by 7:
-2m=70
divide both sides by -2:
m = -35
3(x-0.87)-2x=4.98
distribute the 3 to terms inside parenthesis:
3x-2.61-2x=4.98
combine x's on left:
3x-2.61=4.98
add 2.61 to both sides:
3x=7.59
divide both sides by 3:
x = 2.53


2. Solve each equation by eliminating the fraction
x/2-x/3=5
multiply by a common denominator (6) to get rid of denominators:
6[x/2-x/3] = 6[5]
3x-2x=30
x = 30


1k/15+5=1k/6-10
multiply by a common denominator (30) to get rid of denominators:
30[1k/15+5] = 30[1k/6-10]
2k+150 = 5k-300
150 = 3k-300
450 = 3k
150 = k

0.5b+3.4=0.2b+12.4
subtract 0.2b from both sides:
0.3b+3.4 = 12.4
subtract 3.4 from both sides:
0.3b = 9
b = 30

-6x+3= -7-5x
add 6x to both sides:
3= -7+x
add 7 to both sides:
10 = x
3-3(5-x)=0
distribute the -3 to both terms inside parenthesis:
3-15+3x = 0
-12+3x = 0
3x = 12
x = 4

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I think you meant (please check)

.
If so, finding the vertex gives you your solution.
Axis of symmetry = -b/2a
Axis of symmetry = -30/2(-16)
Axis of symmetry = -30/-32
Axis of symmetry = 15/16
.
Height, then when t = 15/16:

h = 20.06 feet

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a bank offers two checking plans. The anywhere plan charges 30 cents per check. Th Accucheck plan costs $1.12 a month, plus 22 cents per check. For what number of checks will the Accucheck plan costs less than the Anywhere plan?
.
Strategy: determine where the two plans costs the same.
.
Let x = number of checks written
then
.30x = .22x + 1.12
.08x = 1.12
x = 1.12/.08
x = 14
.
Solution:
Accucheck plan costs less when you write MORE than 14 checks per month.

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P=C(1+r/n)^nt for compounding interest what does the ^ stand for and how would you calculate it if:
C= 3150
r=2
n=365
t=1
.

Plugging in our given values:





.
That's $3213.63

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Draw a diagram, it'll help you find a solution.
.
From Pythagorean theorem:
Let b = base of triangle
then
b^2 + 24^2 = 26^
b^2 + 576 = 676
b^2 = 100
b = 10 feet
.
Area of a triangle is:
(1/2)bh
= (1/2)(10)(24)
= (5)(24)
= 120 sq ft
.
Cost:
10 * 120 = $1200

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The length of a rectangle is five times its width. if the perimeter of the rectangle is 72 cm, find its area?
.
Let W = width
then
5W = length (from:"length of a rectangle is five times its width")
.
Since we know the perimeter:
2(W + 5W) = 72
(W + 5W) = 36
6W = 36
W = 6 cm (width)
.
Length:
5W = 5(6) = 30 cm (length)
.
Area:
Width * Length
6 * 30 = 180 square cm

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The second hand of a clock is four inches long. To the nearest inch, how far does the tip of the second hand travel in 45 seconds?
.
If the tip went the entire circle of 60 seconds it would have traveled:
Circumference = (pi)(2r)
where
pi is 3.14
r is radius
.
Circumference = (3.14)(2)(4)
Circumference = (3.14)(8)
Circumference = 25.12
.
If we only traveled for 45 seconds:
45/60 * 25.12
= 3/4 * 25.12
= .75 * 25.12
= 18.84
To the nearest inch:
19 inches

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Apply Pythagorean theorem:
.
Let x = length of one side of square
then





.
Area is then:
x * x
= 5sqrt(2) * 5sqrt(2)
= (5sqrt(2))^2
= 25*2
= 50 square units

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.
x = {-4, 16}
We can throw out the -4 because it won't work therefore:
x = 16

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the length of the diagonal of a square is 80 inches. find the exact dimensions of the square?
.
Let x = length of one side
then applying Pythagorean theorem:







.
Answer: dimension of square:
inches by inches

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log2x=log3 + log(x-4)
log 2x = log 3(x-4)
2x = 3(x-4)
2x = 3x-12
-x = -12
x = 12

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rounded to the nearest tenth, algebraically find the solution(s) of logx+log2x=1
.
log(x * 2x) = 1
log(2x^2) = 1
2x^2 = 10^1
2x^2 = 10
x^2 = 5
x = sqrt(5)
x = 2.2 and -2.2

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Bacteria grow by cell division. If each cell divides into 2 in every 2 minutes how many cells will exist after 32 minutes? Assume there was only one cell at the beginning
.
Apply exponential growth equation:
A(t) = Pe^(rt)
where
A(t) is amount at time t (2)
P is the initial amount (1)
r is rate of growth (unknown)
t is time (2)
.
Finding r:
A(t) = Pe^(rt)
2 = 1e^(2r)
ln(2) = 2r
ln(2)/2 = r
.
Cells at 32 minutes:
A(32) = 1e^(ln(2)/2 * 32)
A(32) = e^(ln(2) * 16)
A(32) = 65536 cells

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ln(8x)=5
e^ln(8x)=e^5
8x = e^5
x = e^5/8
x = 18.552

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a dvd player originally cost $150. It is on sale for $72.50. what is the percent of the discount, rounded to the nearest tenth of the percent?
.
Let x = the discount
then
150 - 150x = 72.50
150(1- x) = 72.50
1-x = 0.48333
-x = -0.5167
x = 0.5167
.
Multiply by 100 to get the percentage:
x = 0.5167*100
x = 51.7%

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A pool can be filled in 12 hours using a pipe alone, or in 28 hours using a hose alone. if water is entering through both the pipe and the hose. How long will it take to fill the pool?
.
Let x = time it takes to fill the pool with both pipes
then
x(1/12 + 1/28) = 1
x(28 + 12) = 336
x(40) = 336
x = 8.4 hours
Or, in hours and minutes
x = 8 hours and (.4 * 60) minutes
x = 8 hours and 24 minutes

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three yeast cells are placed in a laboratory dish at 9:00 am. the number of yeast cells doubles in every 5 minutes period. the number of cells in the dish at 9:30 am the same day is
a) 30 b) 64 c) 192 d) 256 e) 300
.
This is an "exponential growth" problem...apply :
A(t) = Pe^(rt)
.
For your problem, they give:
Let P (initial amount) = 3
then A(t) (amount at time t) = 2(3) = 6
r is growth ratio -- this is what we're looking for
t is 5 minutes
.
So,
A(t) = Pe^(rt)
6 = 3e^(5r)
Dividing both sides by 3:
2 = e^(5r)
ln 2 = 5r
ln(2)/5 = r
.
Finally, for t = 30 (9:30 - 9:00)
we have
A(t) = 3e^(ln(2)/5t)
A(30) = 3e^(ln(2)/5(30))
A(30) = 3e^(ln(2)(6))
A(30) = 3(64)
A(30) = 192 cells

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If Jan can paint a house in 4 hours and Bill can paint the same house in 6, how long will it take for both of them to paint the house together...?
Trying to figure out how to set this one up, it should be simple and was thinking of converting into a fraction with a common denominator... but was sure if that was making it harder than it was ? thanks
Answer given is 2 hrs 24 mins just not sure how 2 get there....
.
Let x = time it takes to paint the house together
then
x(1/4 + 1/6) = 1
Multiplying both sides by a common denominator 24:
24x(1/4 + 1/6) = 24(1)
x(6 + 4) = 24
x(10) = 24
x = 2.4 hours
Or, converting to hours and minutes:
x = 2 hours and (.4*60) mins
x = 2 hours and 24 mins

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Let's imagine that in 2001, a surgeon performs the first brain transplant, the only transplant performed that year. If in 2010, there are 983 brain transplants, find an exponential growth function that fits the data. (Round decimals to three places.)
.
Apply to exponential growth-decline model:
A(t) = Pe^(rt)
where
A(t) is amount after t time
P is the initial amount
r is the rate of growth/decline
t is time
.
Your problem gives you:
A(t) is 983
P is 1
r is what you're looking for
t is 2010-2001 = 9
.
A(t) = Pe^(rt)
983 = 1e^(9r)
983 = e^(9r)
ln(983) = 9r
ln(983)/9 = r
0.766 = r
.
Your "exponential growth function" is:
A(t) = e^(0.766t)

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The floor of a square room is covered in square tiles. If the diagonals of the room have a total of 21 tiles, how many tiles are on the floor?
.
Let x = number of tiles on each side of the square room
then applying Pythagorean theorem:
x^2 + x^2 = 21^2
2x^2 = 441
x^2 = 220.5
x = 14.85 tiles on each side
.
Total number of tiles:
x*x = x^2 = 14.85^2 = 220.5 tiles