# ACT Math: A Challenging Arc Length & Sector Area Problem

One of our more popular posts involves Arc Lengths & Sector Areas for circles. With that in mind, we thought we’d give you a challenging problem (well, really two problems) involving this topic. Give it a try and then scroll down for the answer.

The circle below has its center at O and triangle AOB is equilateral with sides that are each 6 inches.

1.  What is the area, in square inches, of the shaded region?

A.  6π – 3√3
B.  9π – 3√3
C.  6π – 6√3
D.  6π – 9√3
E.  9π – 9√3

# SAT Math: A Challenging Quadratic Function Problem

As you prepare for the SAT, you’ll definitely want to review quadratic functions. You’ll need to know how to solve quadratic equations, as well as understand the properties of the graphs of these functions.

Here’s a challenging problem involving a quadratic function. Give it a try and then scroll down for the solution.

f(x) = a(x + b)(x – 8)

The equation above represents a quadratic function where a and b are constants. If the function is graphed in the xy-plane, its vertex is at the point (2, -54). What is the value of a + b?

(A)  4
(B)  5.5
(C)  6
(D)  7.5

# ACT Math: A Challenging Polygons Problem

We’ve listed this problem under “ACT Math,” but you could just as easily see a problem like this on the SAT Math test as well. Give the problem a try, and then scroll down for the solution.

A regular polygon is covered by a piece of paper so that only one of its interior angles is visible. You are able to determine that the interior angle is four times as large as the exterior angle. How many sides does the polygon have?

A.  8

B.  9
C.  10
D.  11
E.  12

# SAT Math: A Challenging Functions Problem

As my students and I have prepared for the new SAT, one of the things we’ve noticed is that the more challenging Math problems aren’t just computational or problems where you can throw something into your calculator to get the answer. They are more theoretical — you have to really know the math in order to come up with the correct answer.

Let’s consider a challenging problem involving functions.

A function is defined by the equation below.

f(x) = a(x – 2)(x – 10)

If the minimum value of the function is -8, what is the value of a?

A)  -2
B)  1/4
C)  1/2
D)  2

# Another Challenging Shaded Region Problem

We’ve written about shaded region problems before (here and here) and recently offered up a challenging problem (here) on this topic. Here’s another problem that will put your math brain to work. Give it a try!

In the diagram below, segment AB is tangent to the circle at Point A. The measure of ∠OBA is 30 degrees and the circumference of the circle is 8π units. What is the area, to the nearest tenth of a square unit, of the shaded region?

A.  4.8
B.  5.5
C.  5.8
D.  6.0
E.  6.1