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 30 meters so the equation becomes: 2w + 2h = 30.

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

2w + 2h = 30
w + h = \( \frac{30}{2} \)
w + h = 15
w = 15 - h

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

w = 15 - 2w
3w = 15
w = \( \frac{15}{3} \)
w = 5

Since h = 2w that makes h = (2 x 5) = 10 and the area = h x w = 5 x 10 = 50 m²

The area of a circle is given by the formula A = πr² where r is the radius of the circle. The radius of a circle is its diameter divided by two so A = π(\( \frac{d}{2} \))². If the diameter of the logo increases by 50% the radius (and, consequently, the total area) increases by \( \frac{50\text{%}}{2} \) = 25%

To add or subtract terms with exponents, both the base and the exponent must be the same. In this case they are so add the coefficients and retain the base and exponent:

-1a⁴ + 6a⁴
(-1 + 6)a⁴
5a⁴

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 10 = \( \frac{8 \times 10}{100} \) = \( \frac{80}{100} \) = 0.8 errors per hour

So, in an average hour, the machine will produce 10 - 0.8 = 9.2 error free parts.

The machine ran for 24 - 7 = 17 hours yesterday so you would expect that 17 x 9.2 = 156.4 error free parts were produced yesterday.

The commutative property states that, when adding or multiplying numbers, the order in which they're added or multiplied does not matter. For example, 3 + 4 and 4 + 3 give the same result, as do 3 x 4 and 4 x 3.