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Measure the distance of the diagonal (corner to corner) of the screen on your computer monitor to the nearest tenth of a centimeter or six tenths of a inch. Measure the height of the screen the vertical as well. Use the pythagorean theorem to find the horizontal. 13in diagonal and 8 in. high?
.
The "vertical" and the "horizontal" measurements represents the sides while the "diagonal" is the hypotenuse.
.
Let h=horizontal
then
h^2 + 8^2 = 13^2
h^2 + 64 = 169
h^2 = 105
h = 10.25 inches (horizontal length)

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Answer is B
.
Multiply by (3-i)/(3-i):
5/(3+i) * (3-i)/(3-i)
= 5(3-i)/(3+i)(3-i)
= 5(3-i)/(9-3i+3i+1)
= 5(3-i)/(9+1)
= 5(3-i)/10
= (15-5i)/10
= 15/10 - 5i/10
= (3/2) - (1/2)i

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5/ (3 + i) is equivalent to what in a + bi form.
.
Multiply by (3-i)/(3-i):
5/(3+i) * (3-i)/(3-i)
= 5(3-i)/(3+i)(3-i)
= 5(3-i)/(9-3i+3i+1)
= 5(3-i)/(9+1)
= 5(3-i)/10
= (15-5i)/10
= 15/10 - 5i/10
= (3/2) - (1/2)i (this is the form they want)

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.
Your first instinct with FOIL is correct:

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The diagonal of a square is 42.5 cm. Calculate the perimetre and area of the square.
.
Let x = length of one side of the square
then from Pythagorean theorem we have:
x^2 + x^2 = 42.5^2
2x^2 = 1806.25
x^2 = 903.125
x = 30.052
.
Perimeter = 4x = 120.208 cm
Area = x^2 = 903.125 sq cm

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Paolo's monthly salary is 1 3/7 that of Francesca's. Francesca's monthly salary is 7/5 of Dante's. What fraction of Paulo's earnings is Dante's?
.
Let D = Dante's salary
and P = Paolo's monthly salary
then
(7/5)D = Francesca's salary
.
So, Paolo's monthly salary is:
(1+3/7)(7/5)D =P
(7/7+3/7)(7/5)D =P
(10/7)(7/5)D =P
(10/1)(1/5)D =P
(2/1)(1/1)D =P
(2)D =P
D = (1/2)P
.
So, Dante's salary is 1/2 of Paulo's

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Write the standard equation of the circle passing through (2,5) with the center at (-2,-3).
.
Equation of a circle:
(x-h)^2 + (y-k)^2 = r^2
where
(h,k) is the center
r is the radius
.
The problem gives the center at (2,5)
Plugging that into our formula we have:
(x-2)^2 + (y-5)^2 = r^2
The only thing we need is the radius. Get this from the distance between (2,5) and (-2,-3):
d = sqrt[(x2-x1)^2 + (y2-y1)^2]
d = sqrt[(2+2)^2 + (5+3)^2]
d = sqrt[(4)^2 + (8)^2]
d = sqrt[16 + 64]
d = sqrt[100]
d = 10 (this is our radius)
.
so now we finally get:
(x-2)^2 + (y-5)^2 = 10^2

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h = 1 + 8t - 5t^2.
a) find the height of the cork after 0.8s....i have the answer of 4.2m for this.
Correct.
b) find the time when the cork is at ground level (h = 0)......i have the answer of 1.72s for this.
Correct.
c) explain what the maximum value of t is for which this function is a plausible model for the height of the cork?....i don't understand this question.
hope you can help me by explaining how to do this
.
By inspection of:
h = 1 + 8t - 5t^2
Rewritten as:
h = -5t^2 + 8t + 1
.
We see that is is a parabola and it opens downward (upside down U). We can tell by looking at the coefficient associated with the t^2 term (-5) -- since it is negative, think "sad face". Because of this, we now know that the VERTEX of the parabola is your maximum height/time pair.
There are several ways to find this vertex.
One way is to find the "axis of symmetry"
t = -b/2a = -8/(2(-5)) = -8/(-10) = 8/10 = 4/5 = .8 seconds
.
Height, at that time is:
h = -5t^2 + 8t + 1
h = -5(.8)^2 + 8(.8) + 1
h = 4.2m
Which is "part a" of this problem.

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Factor the polynomial expression. Write each factor as a polynomial in descending order.
x2 + 8x + 15
.
(x+5)(x+3)

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A candy shop mixes nuts worth $1.10 per pound with another variety worth $0.80 per pound in order to make 42 lb of a mixture worth $0.90 per pound. How many pounds of each kind of nut should be used?
.
Let x = amount of $1.10 per pound nuts used
then
42-x = amount of $0.80 per pound nuts uses
.
1.10x + .80(42-x) = .90(42)
1.10x + 33.6 - .80x = 37.8
.30x + 33.6 = 37.8
.30x = 4.2
x = 14 pounds ($1.10 per pound nuts)
.
$0.80 nuts:
42-x = 42-14 = 28 pounds ($0.80 per pound nuts)

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the volume of a sphere is given by V=4/3TTR^3where R= the radius of the sphere. If the radius is decreased by 25%, find the change in the volume.
.
Original volume:
V=(4/3)(pi)R^3
.
If radius = R - .25R = (3/4)R
V=(4/3)(pi)[(3/4)R]^3
V=(3/4)^3(4/3)(pi)R^3
V=(27/64)(4/3)(pi)R^3
.
Comparing the original to the one above, we find that the change in volume will be 27/64 of the original volume

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Can you help me solve this question: 1)George is building a porch wrapped around his whole house. The dimensions of his house are 120m by 160m. If his porch is going to be the same area as his house, how wide will the porch be?
.
Drawing a diagram of the problem will help you "see" how to approach it.
.
Let w = width of porch
then
"area of house" = "area of porch
120 * 160 = [(120+2w)(160+2w)] - (120 * 160)
19200 = [(120+2w)(160+2w)] - 19200
38400 = (120+2w)(160+2w)
38400 = 19200+240w+320w+4w^2
0 = -19200+560w+4w^2
0 = 4w^2+560w-19200
0 = w^2+140w-4800
Solve for w using the quadratic equation we get:
w = {28.489, -168.489}
Toss out the negative solution leaves us with:
w = 28.489 meters
.
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=38800 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 28.488578017961, -168.488578017961. Here's your graph:

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I need help factoring 12x squared+19x+4
.
Apply the "ac method":
We get
(3x + 4)(4x + 1)

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Sam is building a cabinet. The materials that he needs cost $480. To determine the price of the cabinet for his customer, Sam charges 30% more than the cost of the materials. Sam asks for a downpayment of 40% of the total before the work begins. The costumer pays the balance when the job is finished.
.
Price of the cabinet (to the customer) is:
"cost of materials" + "30% of cost of materials"
480 + .30(480)
= 480(1+.30)
= 480(1.30)
= $624 (cost of cabinet)
.
Down payment (due at start of work):
.40(624) = $249.60
.
Amount due at completion:
624-249.60 = $374.4

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Begin by multiplying both sides by 5^x

Next, multiply both sides by 10:




Take log base 5 of both sides:

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x^(-2)=16
Is equivalent to:
1/x^2 = 16
1 = 16x^2
1/16 = x^2
sqrt(1/16) = x
+-1/4 = x
(that's plus or minus 1/4)

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1. Log base 8 X=1/3
x = 8^(1/3)
x = 2
.
2. Log e^x^2=125
x^2 = ln(125)
x^2 = 4.8283
x = 2.197
.
3. Solve for X: Log(5-2x)=log (3x+1)
5-2x = 3x+1
5 = 5x+1
4 = 5x
4/5 = x
0.8 = x
.
4. Solve for X: Log(x+1)-Log(x-1)=1
Log[(x+1)/(x-1)]=1
(x+1)/(x-1)= 10^1
(x+1)/(x-1)= 10
(x+1) = 10(x-1)
x+1 = 10x-10
1 = 9x-10
11 = 9x
11/9 = x
1.222 = x

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In a triangle, the first angle measures 3 times the second and the third measures 20 degress less than the second. find each angle's measure.
.
To answer this question, you need to know that the sum of the interior angles of any triangle is 180 degrees.
.
Let x = measure of second angle
then
3x = measure of first angle
3x-20 = measure of third angle
.
x + 3x + 3x-20 = 180
7x-20 = 180
7x = 200
x = 28.57 degrees (second angle)
.
First angle:
3x = 3(28.57) = 85.71 degrees (first angle)
.
Third angle:
3x-20 = 3(28.57)-20 = 65.71 degrees (third angle)

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How do you complete this function equation f(x)=-2x^2+8x-5. I need to find the vertex h,k , compute h and k and graph the quadratic equation. Plus find the max value of f where it occurs.
I started the equation and found h.
a=-2 b=8 c=-5, so using the ormula -b/2a I got -8/2(-2) which gave me -8/-4 which gave me 2. h=2. Where do I go from here?
.
What you found:
-b/2a = 2
is the "axis of symmetry"
.
To find 'k', simply plug 2 back into the equation and solve:
f(x)=-2x^2+8x-5
f(2)=-2(2)^2+8(2)-5
f(x)=-8+16-5
f(x)=3
.
So, (h,k) = (2,3) vertex
.
To plot, looking at the 'a' coefficient (-2), we KNOW it is a parabola that opens downward. Use the "vertex" above and find two additional points to plot. The two additional points could be where the parabola crosses the x-axis:
0=-2x^2+8x-5
.

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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=24 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 0.775255128608411, 3.22474487139159. Here's your graph:

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If the numbers x and y are added to set K {5, 1, 6, 4}, the mean will
increase by 50%. What is the value of xsquare + 2xy+ysquare? ?
.
Calculate "mean" in original set {5, 1, 6, 4}:
(5+1+6+4)/4
= 16/4
= 4
.
From: "the mean will increase by 50%" we get:
(x+y+5+1+6+4)/6 = 4 + .50(4)
(x+y+5+1+6+4)/6 = 4 + 2
(x+y+5+1+6+4)/6 = 6
(x+y+16)/6 = 6
(x+y+16) = 36
x+y = 20
.
If we simply "square" both sides we get:
(x+y)^2 = 20^2
(x+y)(x+y) = 20^2
Applying FOIL:
x^2 + xy + xy + y^2 = 400
x^2 + 2xy + y^2 = 400
.
Solution: 400

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Lauren is shooting at a target at afair. If she hits the target she recieves 50 pences,but if she misses she has to pay 20 pences for the shot. After 15 shots, Lauren finds she has made a profit of 1.20 pounds. How many hits has she had?
.
Let h = number of hits
and m = number of misses
.
Since we have two unknowns, we'll need two equations.
Equation 1 is derived from the fact she took 15 shots:
h + m = 15
.
Equation 2 is derived from the profit of 1.20
(100 pences is equivalent to 1 pound)
(50/100)h - (20/100)m = 1.20
50h - 20m = 120
5h - 2m = 12
.
Using the "substitution method", we solve equation 1 for m:
h + m = 15
m = 15-h
.
We substitute the above into equation 2 and solve for h:
5h - 2m = 12
5h - 2(15-h) = 12
5h - (30-2h) = 12
5h - 30 + 2h = 12
7h - 30 = 12
7h = 42
h = 6
.
Solution: 6 hits

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three goats eat 6 tin cans in 1 hour . how long should it take six goats to eat
3 tin cans ?
.
First determine what ONE goat can do:
6 tins per 1 hour per 3 goats
For 1 goat 2 tins per 1 hour
.
Let t = time it takes
then
6(2/1)t = 3
(12)t = 3
t = 3/12
t = 1/4 hour (or 15 minutes)

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The area of a rectangle is 6, and its diagonal is sqrt{37}. Find its dimensions and perimeter
.
Let x = width
and y = length
.
From Pythagorean theorem
x^2 + y^2 = 37 (equation 1)
.
From the area:
xy = 6 (equation 2)
Solve equation 2 for y:
y = 6/x
.
Substitute aboe into equation 1 and solve for x:
x^2 + y^2 = 37
x^2 + (6/x)^2 = 37
x^2 + (36/x^2) = 37
x^4 + 36 = 37x^2
x^4 - 37x^2 + 36 = 0
(x^2-36)(x^2-1) = 0
x = {1, 6}
.
Therefore, dimensions are
1 feet by 6 feet
Perimeter:
2(6+1) = 14 feet

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Find the volume. Round to the nearest whole number, if necessary.
Cylindrical grain silo with hemispherical top. Where the heighth of the cylinder is 11 ft. and the diameter is 62 ft
.
Volume of cylinder:
(pi)r^2h
= (3.14)31^2*11
= 33192.94 cubic feet
.
Volume of hemisphere:
(1/2)(4/3)(pi)r^3
= (1/2)(4/3)(3.14)31^3
= (4/6)(3.14)31^3
= 62362.49 cubic feet
.
Total = 33192.94 + 62362.49 = 95555.43 = 95555 cubic feet

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Consider the two shapes separately that makes up the tent:
- volume of rectangle -- L*W*H
- volume of triangular prism -- (1/2)BHL
reference:
http://www.mathsteacher.com.au/year9/ch14_measurement/19_prism/prism.htm
.
volume of rectangle:
L*W*H = 15*15*9 = 2025 cubic feet
.
volume of triangle:
(1/2)BHL = (1/2)15*8*15 = 900 cubic feet
.
Total volume = 2025+900 = 2925 cubic feet

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To factor, use "grouping" technique:
x^7-x^6-64x+64
= (x^7-x^6)-(64x-64)
= x^6(x-1)-64(x-1)
= (x^6-64)(x-1)
But wait, we still factor left term:
= (x^3-8)(x^3+8)(x-1)

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Three techniques could be:
1. factoring
2. completing the square
3. quadratic equation
.
1. factoring:
x^2 -10x-39 =0
(x-13)(x+3) = 0
x = {-3, 13}
.
2. completing the square
x^2 -10x-39 =0
x^2 -10x = 39
x^2 -10x+25 = 39+25
(x-5)^2 = 64
x-5 = +-8
x = 5+-8
x = {-3, 13}
.
3. quadratic equation
x^2 -10x-39 =0

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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=256 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 13, -3. Here's your graph:

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Find an equation for which -3 and 4 are solutions.
.
Remember when you "factored" a quadratic and it would give you two solutions?
.
This is sort of a "reverse" process.
If -3 and 4 are solutions, then:
(x+3) and (x-4)
MUST be factors.
.
Now, to find an equation -- multiply the two factors together:
(x+3)(x-4)
FOIL:
x^2 - 4x + 3x - 12
Combine like-terms:
x^2 - x - 12 = 0 (this is what they're looking for)

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A rectangular garden is to be fenced with 100 meters of fencing. The wall of the house will be one side of the garden, so the fencing is needed only on three sides. What is the largest garden possible?
.
Let x = width
and y = length
.
Perimeter:
2x+y = 100
Solving for y:
y = 100-2x
.
Area = xy
Area = x(100-2x)
Area = 100x-2x^2
Area = -2x^2+100x
.
This is a parabola that opens downward -- therefore, finding the vertex gives you the maximum.
axis of symmetry = -b/(2a) = -100/(-4) = 25
.
Largest possible area then is:
Area = -2x^2+100x
Area = -2(25)^2+100(25)
Area = -1250+2500
Area = 1250 square meters

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A rectangular garden is to be fenced with 100 meters of fencing. The wall of the house will be one side of the garden, so the fencing is needed only on three sides. What is the largest garden possible?
.
Let x = width
and y = length
.
Perimeter:
2x+y = 100
Solving for y:
y = 100-2x
.
Area = xy
Area = x(100-2x)
Area = 100x-2x^2
Area = -2x^2+100x
.
This is a parabola that opens downward -- therefore, finding the vertex gives you the maximum.
axis of symmetry = -b/(2a) = -100/(-4) = 25
.
Largest possible area then is:
Area = -2x^2+100x
Area = -2(25)^2+100(25)
Area = -1250+2500
Area = 1250 square meters

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You have a garden with a fence around it and it is rectangular in shape
The perimeter is 500 feet in length and the width is 80% of the length
Find L and W.
.
Draw a diagram of the problem. It will help you see the problem.
.
Let L = length of the rectangular garden
then from "width is 80% of the length"
.80L = width
.
Definition of perimeter for a rectangle:
2(L + .80L) = 500
L + .80L = 250
1.80L = 250
L = 250/1.80
L = 138.889 feet (length)
.
Width:
.80L = .80(138.889) = 111.112 feet (width)
.
These are decimal answers. They may want the answers expressed as fractions.