hectictar

The Triangle Inequality Theorem states that the sum of any 2 sides of a triangle must be greater than the measure of the third side. (Think about it, if one side was the same length as the other two combined that would make a straight line. If it was any longer the other two sides couldn't reach the endpoints.)

This means:

(x) + (x + 4) > (20)

2x + 4 > 20

2x > 5

x > 2.5

It also means:

(x) + (20) > (x + 4)

and

(x + 4) + (20) > x

but the last two don't help any because they are true for all values of x, so it doesn't narrow down the answer any more.

Now you also said the triangle is acute. That means that all angles are less than 90°. Since 20 is the longest side length, the angle across from the side with length 20 is the biggest angle. So x has to be less than whatever values make that angle exactly 90. So just find the values of x when the angle opposite the side with length 20 is 90.

Pythagorean theorem says:

x² + (x+4)² = 20²

x² + (x+4)(x+4) = 400

x² + x² + 8x + 16 = 400

2x² + 8x + 16 = 400

Divide everything by 2.

x² + 4x + 8 = 200

x² + 4x - 192 = 0

Use quadratic formula to figure out that:

x = 12 and x = -16

It can't be negative because we are talking about side lengths.

So x has to be less than 12. x < 12

SO the final answer is 2.5 < x < 12

This key: ÷

(a³)⁴

=(aaa)(aaa)(aaa)(aaa)

= a¹²

The slope is the x-coefficient when the equation is in slope-intercept form.

Slope-intercept form is y =mx + b

The "m" is the slope.

So just get the equation into that form.

2y = 6x - 8

Divide both sides by 2.

y = 3x - 4

Now it is in slope-intercept form. 3 is the x-coefficint, so 3 is the slope.

Press "2nd" then press "atan".

(6y - 11)(6y + 11)

All you need to do is "FOIL" this expression. That means multiply the First two terms  of each factor together, multiply the Outer two terms of each factor together, multiply the Inner two terms of each factor together, and multiply the Last two terms of each factor together. Then add each of those products together.

You can also think of this by saying,"The first term times the third and fourth term plus the second term times the third and fourth term." All this is doing is getting you to multiply each thing by every other thing.

So for this problem that will be:

= (6y)(6y) + (11)(6y) + (-11)(6y) + (-11)(11)

= 36y² + 66y - 66y - 121

= 36y² - 121

Now you can see that a = 36.

Here's how I think of it: How many "15 year periods" are there in 45 years? Once I know how many fifteen year periods are in 45 years, just plug that number in for x into the function.

45/15 = 3

That means there are 3 fifteen year periods in 45 years.

So after 3 fifteen year periods, there will be 75*4³ foxes

That is 75(64) foxes

4,800 foxes

Oh yeah I always forget that thing.

$\int x^{y} dx = \frac{x^{y+1}}{y+1}$

except y = -1

All you do is reverse power rule pretty much.