Will85237

Total distance to cover = 4 * 18 + 2 * sqrt(18^2+18^2) = 123 ft

123 / 25 = 5 rolls

\(\frac{y_2-y_1}{x_2-x_1}=\frac{(9)-(25)}{(2)-(-2)}=\frac{-16}{4}=-4\)

The ladder forms a right-angle triange with the ground and the wall.

Looking at the angle between the ladder and the ground, this means the ladder is the hypotenuse, the wall is the opposite side, and the ground is the adjacent side.

We know the length of the ladder (18 ft) and the height of the wall (17.3 ft), so we know the hypotenuse and opposite side lengths, so we can use sin to calculate what the angle between the ladder and the ground would have to be.

\(\sin(\theta)=\frac{opposite}{hypotenuse}\)

\(\theta=\arcsin(\frac{opposite}{hypotenuse})\)

\(\theta=\arcsin(\frac{17.3}{18})\)

\(\theta=74.0^{\circ}\)

74 is less than 75, so the ladder will be safe.

5 * (5x + 1)

= 5 * 5x + 5 * 1

= 25x + 5

\(ln(\sqrt e)=\frac{1}{2}\)

ln is just log base "e".

\(log_{e}(\sqrt e)=\frac{1}{2}\)

To convert this expression to an exponent, we take the log base, put it to the power of the otherside of the equation, and then put that equal to what was in the logs brackets.

\(e^{\frac{1}{2}}=\sqrt{e}\)

\(\frac{\sqrt{27}\times\sqrt{50}}{\sqrt{54}}\)

Write the numbers as products of primes.

\(=\frac{\sqrt{3\times3\times3}\times\sqrt{2\times5\times5}}{\sqrt{2\times3\times3\times3}}\)

The 2 and 3s on the top and bottom cancel out.

\(=\sqrt{5\times5}\)

\(=5\)

You could measure the circumference and the diameter of a circle and divide the circumference by the diameter to get an estimate for pi.

First, rewrite the equations so y is on it's own.

\(5x+2y=-2\)

\(2y=-2-5x\)

\(y=\frac{-2-5x}{2}\)

\(3x-5y=17.4\)

\(-5y=17.4-3x\)

\(y=\frac{17.4-3x}{-5}\)

Because both equations are equal to y, we can say they are equal to each other and solve for x.

\(\frac{-2-5x}{2}=\frac{17.4-3x}{-5}\)

\(10+25x=34.8-6x\)

\(31x=24.8\)

\(x=0.8\)

Then subsitute this value for x into either of the original equations to find the value of y.

\(y=\frac{-2-5(0.8)}{2}\)

\(y=-3\)

So x = 0.8 and y = -3

We can add on zeros after the decimal part of the number because it has no effect on the non-zero numbers.

So 30813.76 to 3 decimal places would be 30813.760

The equation for the volume of a rectangular prism is:

\(Volume=width\times length\times height\)

If we multiply the width by 2, the length by 3, and the height by 4 in this equation, we have to do the same to the other side so the equation remains true.

\(2\times3\times4\times Volume=(2\times width)\times(3\times length)\times(4\times height)\)

\(24\times Volume=(2\times width)\times(3\times length)\times(4\times height)\)

Therefore the volume will be 24 times it's original value.