Hopefully you were able to find some time and a quiet place this weekend to try our Five Problems for Your Weekend (3/20/15). Below you’ll find the fully worked-out solutions to those problems.
Begin by re-arranging 4x – 8y = 24 into slope-intercept form.
4x – 8y = 24
-8y = -4x + 24
y = 1/2 x – 3
As you may recall from algebra, perpendicular lines have slopes that are opposite reciprocals. Because the slope of the original line is 1/2, we are looking for a line with a slope of -2. That makes the correct answer Choice B.
This is a tricky little problem — it depends upon your understanding of factoring and exponents rules. Start by taking out a Greatest Common Factor (GCF) and then substitute 5 for ab in a couple of places.
a²b² + a³b³
a²b²(1 + ab)
(ab)²(1 + ab)
(5)²(1 + 5)
One of the keys to getting this problem correct is to recognize that a²b² is equivalent to (ab)². You really need to know your exponent rules well.
You’ll need to use Pythagorean Theorem twice to find the lengths of both segment BD and segment DC.
24² + (BD)² = 26²
576 + (BD)² = 676
(BD)² = 100
BD = 10
10² + (DC)² = (√149)²
100 + (DC)² = 149
(DC)² = 49
DC = 7
You can now determine that the length of segment AC, the base of the triangle, is 31. The area of the triangle then is ½(31)(10) = 155 square units.
This is a right triangle trigonometry (SOH-CAH-TOA) problem. The ramp is the hypotenuse of the right triangle and we are looking for the side opposite the given angle, so we can write an equation involving sine.
sin 3.8 = x/20
20(sin 3.8) = x
x ≈ 1.3
For the final calculation, be sure that your calculator is set to degrees and not radians.
Because the tangent of ∠ A is 2/3, you know the ratio of the opposite side to the adjacent side is 2 to 3. Those sides can be labeled 2x and 3x.
You can now use the Pythagorean Theorem to find the value of x.
(2x)² + (3x)² = (√117)²
4x² + 9x² = 117
13x² = 117
x² = 9
x = 3
Because segment CB has been represented by 2x, its length is 2(x) = 2(3) = 6.
If you have questions about these problems or anything else to do with the ACT, leave a comment below or send me an email at firstname.lastname@example.org.