Hopefully you were able to find some time and a quiet place this weekend to try our Five Problems for Your Weekend (3/6/15). Below you’ll find the fully worked-out solutions to those problems.
The two numbers need to satisfy the equation xy + 8 = 44. Subtracting 8 from each side of the equation tells you that the product of x and y must be 36. Of the answer choices, the only one that doesn’t give you that product is B (8 x 4 = 32, not 36).
To solve an absolute value equation, you should write and solve two separate equations. If the absolute value of some quantity is 5, then the quantity must equal either 5 or -5.
x – 4 = 5 or x – 4 = -5
x – 4 + 4 = 5 + 4 x – 4 + 4 = -5 + 4
x = 9 or x = -1
Alternatively, you could have simply tried each of the answer choices to see which one worked. Beware of Choice G, however. Although 9 is a solution, it is not the only solution. After you determined that 9 was a solution, you would need to test any other answer choice that included x = 9.
Let’s first lay out what we have. We know that the third term is 18 and the sixth term is 39. The remaining terms can be represented with blanks.
____ _____ 18 _____ _____ 39
To go from the third term to the sixth term, we would need to add the common difference three times. If we increased 21 units (39 – 18) in three additional terms, the difference between the terms must be 7 (21 ÷ 3). Now we can fill in the remaining terms.
4 7 18 25 32 39
The first term is 4.
Let’s first use the perimeter formula for a rectangle to find the width of the rectangle (which is also the radius of the circle).
2L + 2W = P
2(10) + 2W = 28
20 + 2W = 28
2W = 8
W = 4
Now you can use the fact that the radius of the circle is 4 to find the area of the circle.
A = πr²
A = π(4)²
A = 16π
The shaded portion of the circle is a quarter circle because ∠CAB is an angle of the rectangle and, therefore, 90 degrees. So the shaded region is equal to 1/4(16π) = 4π.
The equation of a circle is (x – h)² + (y – k)² = r², where (h, k) is the center of the circle and r is the radius of the circle. The equation of our circle is (x + 4)² + (y + 2)² = 9, which can be thought of as (x – (-4))² + (y – (-2))² = 3². This tells you that the center of the circle is (-4, -2) and that the radius is 3. You can now sketch the circle in the standard xy-coordinate plane. Doing so will show you that the circle lies in Quadrant II and Quadrant III.
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.