Hopefully you were able to find some time and a quiet place this weekend to try our Five Problems for Your Weekend (4/3/15). Below you’ll find the fully worked-out solutions to those problems.
If 30% of the students chose basketball, then 30% of the circle should be taken up by the pie piece that represents that answer. Remember that a full circle is 360 degrees , so 30% of a circle is 360(.30) = 108 degrees.
The slope of a line is defined as the ratio of the change in y to the change in x. If the slope of our line is -3, that means that y decreases by 3 every time x increases by 1. Now let’s take a look at our points (5, 11) and (7, b). The x-coordinate increased by 2, from 5 to 7. Therefore, the y-coordinate should decrease by 6. The y-coordinate for the second point should be 11 – 6 = 5.
Another possibility would be to write an equation using the slope formula.
From here you can use cross products to solve for b.
1(b – 11) = (-3)(2)
b – 11 = -6
b – 11 + 11 = -6 + 11
b = 5
The distance between the center of the circle and any point on the circle is its radius, which is what we need to know in order to find the area of the circle. Let’s plug the coordinates of the two points into the Distance Formula.
The radius of the circle is √50 and we know that the area of a circle can be found using the formula A = πr². The area of our circle is π(√50)² = 50π.
If, however, you forgot the Distance Formula, you can still get this problem right! Sketch the problem out and use Pythagorean Theorem (Distance Formula is really Pythagorean Theorem “in disguise” … for more on this, see here).
a² + b² = c²
7² + 1² = c²
49 + 1 = c²
50 = c²
√50 = c
Multiplying complex numbers is a lot like doing a “FOIL” problem.
( 4 – i )²
( 4 – i )( 4 – i )
16 – 4i – 4i + i²
16 – 8i + i²
16 – 8i + (-1)
15 – 8i
The important thing to remember is that i² = -1, but that’s given to you in the problem.
A really good point to make here is that you need to be aware of where you are on the test. If you’re on #55 and you think you see a “really easy” solution, that’s probably not it. You’re missing something. In this case, the answer that might draw you in is to average 20 and 30 and choose 25 as your answer. That’s seems way too easy for a problem at this point in the test. Let’s think about it more carefully.
We know that the average speed is going to be the total distance divided by the total time. We could do this abstractly, but this is a great place to try picking numbers. Let’s find a distance that works well with our rates. One possibility is to use 60 miles.
To your friend’s house: 60 miles at 20 miles per hour = 3 hours
Return trip: 60 miles at 30 miles per hour = 2 hours
Total miles: 60 + 60 = 120 miles
Total time: 3 + 2 = 5 hours
Average speed: 120 miles in 5 hours = 120/5 = 24 miles per hour
If you have questions about these problems or anything to do with the ACT, send us an email at firstname.lastname@example.org.