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Tip #50 to Get a Top ACT Math Score (page 3)

By Brian Leaf
McGraw-Hill Professional
Updated on Sep 7, 2011

Hard

1. formula for the surface area of a rectangular solid can be SA = 2lw + 2lh + 2wh where l represents the length of the base, w represents the width of the base, and h represents the height of the solid. By halving each of the dimensions (l, w, and h) of a certain geometric solid, the surface area will be multiplied by what number?
1. 0.1
2. 0.25
3. 0.33
1. 0.50
2. 0.75
2. The dartboard shown below has a small circle, with radius 3.5, inside a larger circle, with radius 5. When Matty randomly throws a dart at the board, which of the following is closest to the probability that it will land in the shaded region?
3. If the width of a little league baseball must satisfy the inequality |w – 3| ≥ 0.12, what is the widest that a ball can be?
1. 2.12
2. 2.88
3. 3.00
1. 3.12
2. 3.15

1. E   Exponents (Skill 30) and avoiding careless errors (Skill 44). 2(3x4)3 = 2(27x12) = 54x12. Careless Error Buster: Remember to apply the exponent to the 3 as well as the x4, and remember to multiply by 2 only after USING the exponent.
2. J   When you have a right triangle, try a2 + b2 = c2 (Skill 24). The legs are 7 and 24, so (7)2 + (24)2 = c2, and 49 + 576 = c2. So 625 = c2, and 25 = c. Remember to plug in the legs (the shorter sides) for a and b and the hypotenuse (the longest side, always opposite the right angle) for c.
3. D   Translation (Skill 27). \$34 equals the original price minus 31% of the original price, so 34 = x – 0.31x.
4. K   Zero Times Anything Equals Zero (Skill 37). You can just "Use the Answers" on this type of question; try the choices and use the process of elimination, eliminating the ones that don't work. That's a quick, reliable, and easy way to get this one. Or you can factor x2 – 5x – 14 = 0 to (x – 7)(x + 2) - 0. Since zero times anything is zero, either x – 7 = 0 or x + 2 = 0. Solve each of these to get x = 7 or –2.
5. A   Midpoint formula (Skill 17). Midpoint is really just the average of the x's and the average of the y's. Just plug in what you know and solve for the variable:
6. In other words, . When you see a proportion, cross–multiply, and solve to get x = –8 and y = 7. (–8, 7). You could also get this question, of course, by "Using the Answers," just trying the choices to see which one works.

7. F   Trig (Skill 34). When a picture is described, draw it. That always helps. Use SohCahToa; the smallest side is 6 and the hypotenuse is 10, so sin
8. A   Weird Number Behavior (Skill 46) and "Make It Real" (Skill 18). Without our Skills many kids would get this one wrong. I almost did, just now, as I was writing this solution. It's tempting to say, "Oh, easy, 9m must be the biggest." But remember our Weird Number Behavior Skill, the higher the digit of a negative number, the smaller the number actually is. Let's "Make It Real," and you'll see what I mean. Let's say m = –3. Now –3 is greater than 9(–3), which equals –27.
9. G   Similar triangles (Skill 26) and special right triangles (Skill 25). As soon as you see a right triangle with a 30 or 60 degree angle, consider special right triangles. A 30–60–90 triangle has sides that fit the pattern x, x , 2x. So since triangle PQR has a 60 degree angle and a side of 3, it must have sides of 3, 3 6. And triangle MNO, since it is similar, will have sides that are proportional to 3, 3, 6. Choice G works since 6, 6 and 12 is proportional (double) to 3, 3, 6
10. E   Translation (Skill 27). The question translates to: 20 – 4p = a negative. You could also say 20 – 4p < 0. Solve this to get
11. Remember to flip the > sign when you divide by a negative! As is so often the case, you could also just "Use the Answers" to find which one works. Choices A, B, and D do not work. Choice C works, but 20 is not the only solution; when you test choice E, with any number more than 5, it also works, so choice E is correct.

12. G   y = x2 + bx + c (Skill 38). For the equation y = x2 + bx + c, the sign in front of the x2 term tells us if the U– shaped graph opens up or down, and the c term tells us the y intercept of the graph. The graph shown in the question opens up, so it should not have a negative in front of the x2; and it has a negative y intercept, so c should be negative. It also must have an x2 since it is a parabola (a U–shaped graph). That eliminates choices F, J, and K. Choices G and H are left. Choice G is correct since the –1 outside the parenthesis represents the y intercept, whereas the –1 inside the parenthesis relates to a shift of the x values, not y values. If that last part is confusing, you could also just graph the two choices on your calculator and see which one matches the diagram shown in the question.
13. D   Algebra (Skill 2) and avoid careless errors (Skill 44). Simplify the inequality:
14. 3(3x – 2) < 6 – (5x – 2)

9x – 6 < 6 – 5x + 2

14x < 14

x < 1

Careless Error Buster: Remember to distribute the negative sign.

Once again, you could also just "Use the Answers" and see which choice works.

15. G   Make It Real" (Skill 18) and area/volume (Skill 19). Let's say l = 4, h = 6, and w = 8; then the surface area of the solid is SA = 2lw + 2lh + 2wh = 2(4)(8) + 2(4)(6) + 2(8)(6) = 64 + 48 + 96 = 208. When we halve each dimension, we get l = 2, h = 3, and w = 4, so the new surface area is SA = 2lw + 2lh + 2wh = 2(2)(4) + 2(2)(3) + 2(4)(3) = 16 + 12 + 24 = 52. So the new surface area is onequarter the original; the original times 0.25 equals the new surface area. Without "Make It Real" many people would incorrectly pick choice J, since we halved the dimensions.
16. C   Probability (Skill 36), donut area (Skill 20), or use the diagram (Skill 23). Probability = , so the probability of hitting the shaded region is
17. You could also use the diagram and realize that the shaded region being half the area of the larger circle is the only answer choice that makes sense with the diagram.

18. J   Absolute value (Skill 41) and "Use the Answers" (Skill 1). You could solve the inequality statement or just try the choices and see which is the greatest one that works. Choices F, G, H and J work, but choice J is the largest one that works.

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