# Hard Math Problem 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.

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is written on the first card, on the second, on the third and so on through , with no numbers repeated. What is the probability that the number on the card is an irrational number? In order for a polynomial to be divisible by a linear binomial, the polynomial and the binomial must have the same root.

The Factor Theorem states that a polynomial P(x) is divisible by binomial (x – c) if and only if P(c) = 0.

Instead, it’s because they take some of the foundations of math topics you’ve already studied in school, like pre-algebra, algebra, and geometry, and turn them into multi-step processes that may combine concepts from different areas. For the middle part of the trip, we know that D = 10mph x 3 hours, so we know that D = 30 miles. We have to know how low the lowest values in this region are. We know that = – Now, if you happen to have the decimal approximations memorized, you will see that: = – If you don’t have this decimal equivalent memorized, you could think about the triangles in QIV: Even if we can’t directly compare the sizes of the two vertical legs, we definitely can compare the two horizontal legs, and 0.8 is definitely bigger than 0.5.

The shaded region inside the circle and outside the triangle has an area of square centimeters. Now that we have the common ratio, we can move from term to term. Suzanne drove for 2 hours 3 hours so the Total Time was 5 hours. If we had just averaged the two speeds (10mph and 20mph) we would have gotten 15mph. Since Suzanne spent more time in the problem going 10mph than 20mph, it makes sense that the Average Speed would be closer to 10mph. ANSWER: D To find the average speed of the bus, we know we will need to find the Total Distance and the Total Time, so we can start by using another formula (Distance = Rate x Time) to help us find the pieces we’re missing for each part of the trip. Sine is positive in QII, zero at the negative x-axis, and starts to get more and more negative as we go around through QIII. = – = This does not go down as far as Here’s a diagram with region (D) and the line = – . This leaves (E) by the process of elimination, but let’s verify that this works. The sine is negative one at the bottom, where the unit circle intersects the negative y-axis.

This definitely eliminates B, and thinking logically, you could probably take a guess that the overall probability is going to be high. ANSWER: C This question is difficult primarily because you need to have some higher-level knowledge of the equations of circles and ellipses.

You should find that out of the first 11, 8 numbers are irrational.

This is a useful trick for problems like this on the ACT.

What’s important is that you visualize what an ellipse like this looks like: If we put a circle inside this shape, it can only have a diameter that is as wide as the ellipse is wide, otherwise it wouldn’t fit. Back to the ellipse equation: , or 9, in our answer choice for the radius of the circle, making our answer C. ANSWER: A Probably the most elegant way to solve this problem is to remember the Factor Theorem.

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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