# Hard Math Problem

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If the students arrived back at Thomas Jefferson High School two hours later, approximately what was the average speed for the entire field trip? The moment you find an irrational number as you count through the series, you can eliminate answer choice A, for example.

Finally, the bus drove 40 miles straight back to the high school. So we just need to count the perfect squares before 50. So that means out of our 50 numbers, 7 can be reduced to integers (or fractions) and 43 are irrational. Bonus hint: If you come across a problem like this on the ACT, and don’t know how to solve it, make sure you eliminate some answer choices!

The function values for p(x) vary directly as x for all real numbers. This information helps us find the fraction of the circle delimited by the triangle.

This means that the area of the unshaded sector of the circle inside the triangle must be , since these two regions must add up to the total area of the circle: .

We multiply by the common ratio to change any one term into its successor.

P(x) = x s(-2) 6 = 0 -8 5*4 – 2s 6 = 0 -8 20 6 = 2s 18 = 2s 9 = s So our answer is A. ANSWER: D In a geometric series, each term is the product of previous term times some common ratio. This means that AD is the radius and should be half of 22 cm, or 11 cm. In the case of this question, we know that AC is 22 cm, that D is the midpoint of AC, and that D serves as a point of intersection between the circle and the triangle. This means that all of the details in the question give you important clues that you need to solve the problem. ANSWER: C There are two ways to solve this problem: the “math” way and the “test prep strategy” way. You see, this problem is a great candidate for plugging in numbers for . ANSWER: E The ACT rarely gives you any unnecessary information in a math word problem. In our problem, we have an ellipse that is taller than it is wide. Which of the following is an equation of the largest circle that can be inscribed in the ellipse with the following equation? The a always goes with the variable whose axis parallels the wider direction of the ellipse, and the b always goes with the variable whose axis parallels the narrower direction, hence the reason for the difference in the equations. In other words, x = -2 must be a root of the equation. If we want polynomial P(x) to be divisible by (x 2), it must be true that P(– 2) = 0. So: We are told in the problem that the area of the shaded region is . Knowing that 11 cm is the radius allows us to find the area of the entire circle using the equation .

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