The hourly error rate for this machine is the error rate in parts per 100 multiplied by the number of parts produced per hour:

\( \frac{8}{100} \) x 5 = \( \frac{8 \times 5}{100} \) = \( \frac{40}{100} \) = 0.4 errors per hour

So, in an average hour, the machine will produce 5 - 0.4 = 4.6 error free parts.

The machine ran for 24 - 2 = 22 hours yesterday so you would expect that 22 x 4.6 = 101.2 error free parts were produced yesterday.

The area of a rectangle is width (w) x height (h). In this problem we know that the rectangle is twice as long as it is wide so h = 2w. The perimeter of a rectangle is 2w + 2h and we know that the perimeter of this rectangle is 54 meters so the equation becomes: 2w + 2h = 54.

Putting these two equations together and solving for width (w):

2w + 2h = 54
w + h = \( \frac{54}{2} \)
w + h = 27
w = 27 - h

From the question we know that h = 2w so substituting 2w for h gives us:

w = 27 - 2w
3w = 27
w = \( \frac{27}{3} \)
w = 9

Since h = 2w that makes h = (2 x 9) = 18 and the area = h x w = 9 x 18 = 162 m²

In order to find how many additional workers are needed to staff the extra crews you first need to calculate how many workers are on a crew. There are 15 workers at the company now and that's enough to staff 5 crews so there are \( \frac{15}{5} \) = 3 workers on a crew. 8 crews are needed for the busy season which, at 3 workers per crew, means that the roofing company will need 8 x 3 = 24 total workers to staff the crews during the busy season. The company already employs 15 workers so they need to add 24 - 15 = 9 new staff for the busy season.

48% of the water consumption is industrial use and 16% is personal use so (48% - 16%) = 32% more water is used for industrial purposes. 25,000 gallons are consumed daily so industry consumes \( \frac{32}{100} \) x 25,000 gallons = 8,000 gallons.

To multiply terms with exponents, the base of both exponents must be the same. In this case they are so multiply the coefficients and add the exponents:

-2a⁵ x 5a⁵
(-2 x 5)a^{(5 + 5)}
-10a¹⁰