# ACT Math: Five Solutions (5/8/15)

Hopefully you were able to find some time and a quiet place this weekend to try our Five Problems for Your Weekend (5/8/15).  Below you’ll find the fully worked-out solutions to those problems.

9.  C
You can simplify the polynomial expression by multiplying using FOIL and combining like terms.

(4x + 9)(4x – 9)
16x² – 36x + 36x – 81
16x² – 81

24.  G
The cosine of an angle is the ratio of the length of its adjacent side to the length of its hypotenuse. For angle θ, the adjacent side has length a and the hypotenuse length c. Therefore, cos θ = a/c.

35.  C
For this problem, you can set up an equation using the slope formula and solve for k.

$m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$
$\frac{1}{2}=\frac{k-2}{1-(-5)}$
$\frac{1}{2}=\frac{k-2}{6}$
$6=2(k-2)$
$6=2k-4$
$10=2k$
$5=k$

42.  G
You can approach this problem by picking a number for the regular price of the sweater. For a percent problem, a good number to choose is 100 because percent literally means “per hundred.” Once you’ve chosen this price, the calculations look like this:

Sale Price:  100 – .30(100) = 100 – 30 = 70
Employee Discount: 70 – .20(70) = 70 – 14 = 56

Sarah pays 56% of the regular price of the sweater.

55.
The amplitude of a periodic function is the distance between its horizontal midline and the highest (or lowest) point of the function. You can also compute it by finding one-half the distance between the highest and lowest points. For this function, the amplitude is 3.

The period of the function is the horizontal distance that it takes to complete one instance of the repeating pattern. Picking any starting point (x = 0 is a good choice) and trace the function until the pattern is complete and it starts to repeat itself. The period for this function is 2.

So a + p = 3 + 2 = 5.

If you have questions about these solutions or anything to do with the ACT, send us an email at info@cardinalec.