Anneliese's Responses in WyzAnt Answers

A five paragraph essay will typically have the following:

Introduction paragraph (¶1): state your thesis (your main point) and briefly outline each of your supporting points.

Supporting paragraphs (¶2-4): each will have a topic sentence that states the main theme of the paragraph, followed by evidence supporting that idea. Of the three supporting paragraphs, each should have its own distinct idea and evidence which supports your overall thesis.

Conclusion paragraph (¶5): restate/summarize the evidence that you have presented, leading up to a final statement of your thesis.

Picture this as a right triangle. The "distance from the wall" (i.e. the ground) is the bottom of the triangle, and the ladder is the hypotenuse. You are given the height as 6 ft, so this is the third side of the triangle.

Let x equal the ground / base of the triangle. 

Because this is a right triangle, we can use the Pythagorean theorem (a² + b² = c²), where a and b are the two sides making the right angle, and c is the hypotenuse.

We are given in the problem that x is two feet more than the ladder / hypotenuse. So c = x+2. One side of triangle can be a, and the other b. Let's say that a is the height, which is given as 6, and b is the ground, which we are calling x:

a=6                       b=x                c=x+2

Plug this into the Pythagorean theorem:    6² + x² = (x+2)²

First, simplify the right side by expanding it:
(x+2)² = (x+2)(x+2) = x² + 4x + 4
6² + x² = 36 + x² = x² + 4x + 4

Subtract 4 from both sides: 32 + x² = x² + 4x

Subtract x² from both sides: 32 = 4x

Divide both sides by 4: 8 = x

Remember that x was equal to the distance from the wall, and the length of the ladder is two feet more than x.

Therefore, the length of the ladder = x+2 = 8+2 = 10 feet.

First, you need to factor the equation. Group 2x³ and 8x together, and x² and 4 together:

(2x³ + 8x) + (x² + 4)

Factor the first part of the equation:
2x(x² + 4) + (x² + 4)

Notice that we now have the same thing in parenthesis twice. We can think of this as:
2x(x² + 4) + 1(x² + 4) = (2x + 1)(x² + 4)

By the Zero Product Property, our zeros will come from:
2x + 1 = 0
and
x² + 4 = 0

2x+1=0               subtract 1 from both sides, then divide by 2
x = -1/2

x² + 4 = 0           subtract 4 from both sides, then take the square root. note that we will have to use i since we are taking the square root of a negative number.
x = 2i

I'm assuming you want to solve for x. 

You want to get everything over a common denominator. In the case of 5 and 7, the lowest common denominator is 35.

To x/7 to have a denominator of 35, we have to multiply it by 5/5: (x/7)*(5/5) = 5x/35

Do the same with 5/7: (5/7)*(5/5) = 25/35

With x/5, to get the denominator to 35 we multiply by 7/7: (x/5)*(7/7) = 7x/35

Now we have: 5x/35 = 7x/35 + 25/35. Multiply both sides by 35 to eliminate all the denominators:
(5x/35 = 7x/35 + 25/35)*35 ==> 5x = 7x + 25

Solve for x by combining like terms. Subtract 7x from both sides: -2x = 25

Divide both sides by -2: x = -12.5

Let's say x is the amount invested at 11%, and y is the amount invested at 14%. We know that the two together total 22,000, so:
x+y = 22,000
y = 22,000 - x

Now write an equation representing the information given in the problem, remembering that 11% = 0.11, and 14% = 0.14:
0.11x + 0.14y = 2810
0.11x + 0.14(22,000 - x) = 2810

Expand the equation:
0.11x + 3080 - 0.14x = 2810                 combine like terms
-0.03x + 3080 = 2810                           subtract 3080 from both sides
-0.03x = -270                                       divide both sides by -0.03
x = 9000

Solve for y:
y = 22,000 - x = 22,000 - 9000 = 13,000

So the amount invested at 11% is $9000, and the amount invested at 14% is $13,000