Hopefully you were able to find some time and a quiet place this weekend to try our Five Problems for Your Weekend (4/24/14). Below you’ll find the fully worked-out solutions to those problems.
You need to identify the pattern in order to determine the next term. The pattern in this case is “add two, add three, add four …. “. To get the next term, add 6 to 21 to get 27.
An isosceles triangle is a triangle that has two congruent sides. The third side of the triangle could be either 3 or 5. (Note: You should also check that the two solutions give you triangles that are possible using the triangle side length inequality: sum of the two smallest sides > largest side. For example, if the two given sides had been 3 and 7, then 3 would not have been a possible third side because 3 + 3 < 7. You can’t make a triangle with sides 3, 3 and 7.)
You can begin by getting expressions for the surface area and the volume of the cube using their respective formulas.
Surface Area = 6s² = 6k²
Volume = s³ = k³
The ratio of the surface area to the volume is, therefore, 6k² : k³. This can be simplified to 6 : k by dividing each of the terms in the ratio by k².
Because the trapezoid is isosceles, you can drop a pair of segments and “break up” the bottom side of the trapezoid as shown in the diagram below. You can then use what you know about 30-60-90 triangles to find the height of the trapezoid. (Note: As soon as you saw the 60º angle in the problem, you probably could have guessed that you’d need to use that special right triangle at some point!)
Now go ahead and compute area of the trapezoid using the area formula.
A = ½(b1 + b2)h
A = ½(6 + 10)(2√3)
A = ½(16)(2√3)
A = 16√3
Also notice that if you had forgotten the area formula for a trapezoid, you could have found the areas of the two triangles and the rectangle and added them.
The first thing you should consider is the sign of cos θ. Because cosine is negative, you know that the angle has to be in Quadrant II (π/2 < θ < π) or Quadrant III (π < θ < 3π/2). That eliminates answer choices A, B, and E.
Now let’s consider the other two choices. It would be helpful to know the cosines of the angles in those answer choices.
cos π/2 = 0
cos 2π/3 = -0.5
cos π = -1
The cosine of our angle (-0.674) is between -.05 and -1 so it stands to reason that our angle could be between 2π/3 and π.
If you have questions about these problems or anything to do with the ACT, send us an email at firstname.lastname@example.org.