ACT Math: Five Solutions (2/13/15)

Hopefully you were able to find some time and a quiet place this weekend to try our Five Problems for Your Weekend (2/13/15). Below you’ll find the fully worked-out solutions to those problems.

11.  C
Let’s choose the variable w to represent the number of weeks. Now write an equation and solve it.

120 + 10w = 210 – 5w
120 + 15 = 210    ← Add 5w to each side
15w = 90    ← Subtract 120 from each side
w = 6    ←  Divide both sides by 15

They will have the same amount of money in six weeks.

28.  F
To find a median you need to place the numbers in order. Because there are seven numbers, the median will be the fourth number. In this case, if the median is 8, the data when listed in order must look like this:

___  ___  ___  8  ___  ___  ___

Because the data already contains three numbers (2, 4 and 7) that are less than 8, Brendan’s number must be greater than 8.  All of the answer choices are possible except 6. In that case the data in order would look like this and the median would be 7.

2   4   6   7   8   12   21

39. C
Begin by making a sketch of the circle. Because the endpoints of the diameter of the circle are (3, 6) and (11, 6), the center of the circle is at (7, 6), the midpoint of the diameter. The diameter has a length of 8, the distance from (3, 6) to (11, 6). Five Solutions 2-13-3Therefore the radius is 4. Finally, if the center is at (7, 6) and the radius is 4, the minimum y-coordinate of the circle is 2, and the circle does not cross the x-axis.

Now look at the three statements about the circle. The center is at (7, 6), so the first statement is true. The radius is 4, so the second statement is also true. However, the circle does not cross the x-axis, so the third statement is not true.

Because only Statements I and II are true, the correct answer is Choice (C).

48. J
Begin by finding the distance between (7, 9), the center of the circle, and (13, 17), the point on the circle. This distance is the length of a radius of the circle. You can do this using the Distance Formula.

d = \sqrt{(y_{2}-y_{1})^{2}+(x_{2}-x_{1})^{2}}
d = \sqrt{(17-9)^{2}+(13-7)^{2}}
d = \sqrt{8^{2}+ 6^{2}}
d = \sqrt{64 + 36}
d = \sqrt{100}
d = 10

If you didn’t remember the Distance Formula, you can still get the problem correct by sketching it out, drawing a right triangle and using the Pythagorean Theorem (the Distance Formula is really Pythagorean Theorem “in disguise”).

Five Solutions 2-13-4

a^{2} + b^{2} = c^{^{2}}
6^{2} + 8^{2} = c^{^{2}}
36 + 64 = c^{^{2}}
100 = c^{^{2}}
10 = c

You’ll get the same answer using either the Distance Formula or by using Pythagorean Theorem. Use the method that makes the most sense to you.

Now that you have the radius, you can compute the area of the circle using the formula A = πr². The area of the circle is π(10)² = 100π.

57.  B
Begin by sketching the triangle associated with this problem. You can use the Pythagorean Five Solutions 2-13-5Theorem to find the missing third side of the triangle.

a^{2} + b^{2} = c^{^{2}}
12^{2} + 5^{2} = c^{^{2}}
144 + 25 = c^{^{2}}
169 = c^{^{2}}
13 = c

As you label the triangle be sure to pay attention to the signs of the sides. The vertical side of the triangle should be labeled with a negative value.

From your knowledge of SOHCAHTOA, you know that the sine of an angle is the ratio of the opposite side to the hypotenuse. In this case that is equal to -5/13.

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 info@cardinalec.com .

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