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Substitute x = 50


round to the nearest million...
Million, approximately.
Substitute x = 60


Round to the nearest million...
Million, approximately.

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Simplify:
1)
The technique to use here is to factor the quantity under the radical (the radicand) looking for perfect squares in the factors. When you do find a factor that is a perfect square, take its square root and move the result outside of the radical.
Like this:
the 4 is a perfect square and its square root is 2, so you can rewrite this as:
=
See the idea?
Simplifying, we get:

2) Simplify:
=

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Start with the formula for the area of a rectangle:

The area, A, is given as:
The width is given as:
Find L:

Factor an x from the binomial on the left side then divide both sides by x.
Divide by x.

The length is represented by the binomial

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The diagonal of the stage(10m) represents the hypotenuse of a right triangle whose legs are the width (W) and the length (L) of the rectangle.
But the problem states that the length (L) is 2 meters longer than the width (W), so the length can be expressed as:
L = W+2.
Now we can apply the Pythagorean theorem () to solve this problem since we are dealing with a right triangle.
c is the hypotenuse of the right triangle, or the diagonal of the rectangular stage which is given as 10m.
a and b are the width of the stage (W) and the length of the stage (L = W+2).
Make the appropriate substitutions into the Pythagorean formula to get:
Substitute L = W+2


Divide through by 2 to simplify a bit.
Subtract 50 from both sides.
Solve this quadratic equation by factoring.
Apply the zero product principle.
or
If then
If then Discard this solution as the width, W, must be a positive value.

The width is 6 meters.


The length is 8 meters.
Check:
Substitute c = 10, a = 8, and b = 6.


OK!

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Let's start by assigning variables to the unknown quantities.
Let x = the required number of gallons of grape juice at $2.25 per gallon.
Now, because we want to make a total of 20 gallons of cranberry/grape juice mix, we will need to add (20-x) gallons of cranberry juice at $1.75 per gallon.
At the end we'll have (hopefully) the required 20 gallons of cranberry/grape juice mix at $2.10 per gallon.
Now we can write the equation that will tell us how much of each we will need.
x($2.25)+(20-x)($1.75) = 20($2.10) Simplify and solve for x.
2.25x + 35 - 1.75x = 42 Combine like-terms.
0.5x+35 = 42 Subtract 35 from both sides.
0.5x = 7 Divide both sides by 0.5
x = 14
So we'll need 14 gallons of grape juice @ $2.25 per gallon plus (20-x) 6 gallons of cranberry juice @ $1.75 per gallon to obtain 20 gallons of cranberry/grape juice mix @ $2.10 per gallon.

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First, you want to write the new equation in the "slope-intercept" form (y = mx+b) unless instructed otherwise.
Second, recall that parallel lines have identical slopes, so if you know the slope of the line whose equation is given (y = -x+3), then you autmatically know the slope of any line parallel to this.
Compare the given equation which is already in the slope-intercept form with the general form:
y = -x+3
y = mx+b
You can see that the slope, m, of the given equation is -1.
So the slope of the new equation will be -1 because the new line is parallel to the given line.
Ok, now you can write the slope (m = -1) into the general equation:
y = -x+b
Now you need to find the value of b, the y-intercept.
You do this by substituting the x and y of the above equation with the x and y values taken from the given point (2, 4) through which the new line passes.
y = -x+b Substitute x = 2 and y = 4
4 = -2+b Add 2 to both sides.
6 = b This is the y-intercept of the new line so now you can write the final equation:
y = -x+6
I hope this clarifies the problem for you!

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Ok, the cost of the frozen yogurt is constant at $2.00.
The cost of the toppings varies and depends on the number of toppings you select, so let's call this variable, x, and for every topping you choose, it will cost $0.50
So you can express the total cost (C) of a cup of frozen yogurt as the sum of the fixed cost ($2.00) of the frozen yogurt plus the variable cost (x dollars) of the number (x) of toppings at $0.50 each.
C = $2.00 + x($0.50)

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The volume, V, of a right circular cylinder is proportional to the square of its radius, r, can be expressed as follows:
where k is the proportionality constant.

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Try this:
(I hope I got this right): Subtract from both sides.
Now divide both sides by -6.

Now you are looking for the angle theta whose cosine is or:
or

degrees.
In radians: Radians.
Check:(Degrees)
Substitute



Check: (Radians)
Substitute

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What are the factors of 120?
In other words, what numbers (integers) will divide into 120 without a remainder?
120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120

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Start with the formula for the perimeter of a rectangle:

You are given that the width (W) is 3 meters less than the length, L, so W = L-3. You are also given the perimeter, P = 14 meters. Making the appropriate substitutions into the formula, you'll get:
Simplify and solve for L
Combine like-terms.
Add 6 to both sides.
Divide both sides by 4.
The length is 5 meters.


meters.
The width is 2 meters.
Check:



meters.

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a) When (at what time, t) will the flare be at 150m (h = 150).
Start with the given formula for the height (h) as a function of time (t):
Now substitute h = 150 and solve for t using the quadratic formula:
The a, b, and c, in the formula come from the standard form of the quadratic equation:
Subtract 150 from both sides.
Simplify.
Solve this quadratic using the quadratic formula:




or approx. 1.7 seconds.

or approx. 17.1 seconds.
b) To find out how long the flare was AT OR ABOVE 150m, just subtract the two times:
= seconds.

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You can work this out!
The sum of the three angles in any triangle is 180 degrees.
If you know that two of the angles add up to 62.5 degrees, then the measure of the third angle must be...?
180 degrees - 62.5 degrees =

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What equation???
What you have here is an expression, and expressions can't be "solved" but they can be simplified, or, given enough data, they can be evaluated.
Factor the trinomial:

The factors will look like:
where: and
It's not too hard to see that if and , then
= and =, so, the factors of the trinomial are:

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First, one small correction!
This is not an "equation" but an "expression" because there is no = sign.
In an equation, you would have something on the left side equal to something on the right side.
Ok, Add:
First, group the like-term:
Perform the indicated addition/subtraction:
Finally...

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Solve for x:
First, apply the "distributive property"
Now subtract 3x from both sides.
Then add 10 to both sides.
Finally, divide both sides by 2.

Check:
Substitute x = 11.
Simplify.

OK!

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Solve for x:
Raise both sides to the 2/3 power.

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Let x = the first odd integer, then x+2 will be the next consecutive odd integer and x+4 will be the next consecutive odd integer after that.
The sum of these is equal to 117, so you can write the equation:
Combine like-terms.
Subtract 6 from both sides.
Finally, divide both sides by 3.
This is the first odd integer.
This is the next consecutive odd integer.
This is the next consecutive odd integer.
Check"
OK!

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Congratulations, your answer is correct. Don't forget the dimensions.
W = 30 feet, L = 60 feet.

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Given the area (A) of a triangular sail: sq.m. and its height (h): m., find the length of its base (b).
Start with the formula for the area of a triangle:
Make the approriate substitutions from the given data of A and h:
Multiply both sides by 2.
Now divide both sides by (x+3).
You could use synthetic division or regular long divion, but either way, you get:

The length of the base is m.

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Given:
and , find:
this can also be written:



, so...

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Well, I don't know about a formula, but here's one way to go about it! Find the daily rate of building by the 6 men.
If it takes 6 men to build a houes in three days, then the 6 men can build 1/3 of the house in one day! So you could write:
6 men = 1/3 house/day. But you want to know how many men it would take to build the house in 1 day, so multiply both sides by 3.
3(6 men) = 3(1/3 house)/day.
18 men = 1 house/day.

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Why not reduce ?
Because it can not be reduced. Why, because the numerator and the denominator have no common factor (besides 1).
The factors of 5 ( a prime number) are 1 and 5.
The factors of 66 ( a composite number) are 1, 2, 3, 6, 11, 22, 33, and 66
So you see, there is no common factor that can be cancelled!

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Given:

Find:
Substitute (5+h) for x above:
Simplify.
Combine like-terms.

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The property of real numbers demonstrated by:
12 + 1 = 1 + 12 is called "The commutative property of addition"
The order in which numbers are added is not important, the result is the same.

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Solve for d:
Square both sides.
Subtract d from both sides.
Subtract 4 from both sides.
Divide both sides by -4
Finally, square both sides.

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The problem can be solved by letting s (the distance in meters) = 19 and solving for t (the time in seconds):
Substitute s = 19:
Subtract 19 from both sides.
Solve this quadratic equation using the quadratic formula: and the a, b, and c, come from
So here, a = 2, b = -3, and c = -19.
Make the substitutions:



or
or
Only the positive answer is meaningful here, so...
It will take 3.9 seconds for the object to move 19 meters.

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Solve for x:
You can always solve a quadratic equation using the quadratic formula:
and, of course, the a, b, and c come from the standard form of the equation:

So, in this case, a = 1, b = 9, and c = 14
Making the appropriate substitutions into the quadratic formula, we get:



or
or
If the given trinomial can be factored, this offers another way to solve the problem:
Factor the trinomial.
Apply the zero product principle.
or , then...
If then
If then
So in both cases, you get the same solutions:


A third way would be to graph the equation () and identify the x-intercepts (also called "the zeros").

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Let M = McGoo's present age and G = Gomer's present age.
From the problem, you can write:
1) M = 8G "McGoo is 8 times as old as his son, Gomer"
2) M+3 = 5(G+3) "In three years, he (M) will be five times his son's (G's) age"
Substitute the M = 8G from equation 1) into equation 2) and solve for G.
2a) (8G)+3 = 5(G+3) Simplify.
8G+3 = 5G+15 Subtract 5G from both sides.
3G+3 = 15 Subtract 3 from both sides.
3G = 12 Finally, divide both sides by 3.
G = 4
Gomer is 4 years old.

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Try this:
Let x = the first even number, the x+2 = the next consecutive even number.
From the problem description, you can write:
Simplify this.

Subtract 452 from both sides.
Factor out a 2 to ease the calculations a bit.
so that:
Solve this quadratic by factoring:
so then...
or
Now since the problem did not restrict the answers to positive integers, there are two solutions:
1) and
2) and

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Find the x-intercept:
The x-intercepts (there will be two of them) occur at y = 0, so...
Solve by factoring:
Apply the zero product principle:
or so then...
If then or...
If then
The x-intercepts are:
and
Check the graph: