Hopefully you were able to find some time and a quiet place this weekend to try our Five Problems for Your Weekend (4/10/15). Below you’ll find the fully worked-out solutions to those problems.
You could do this in a number of ways. Probably the most straightforward way is to list out the first few multiples of each number until you find one the numbers have in common.
24: 24, 48, 72, 96, 120, 144, …
40: 40, 80, 120, 160, 200, …
60: 60, 120, 180, 240, 300, …
Recall that perpendicular lines have opposite reciprocal slopes. You can begin by rearranging the given line into slope intercept (y = mx + b) form.
6x – 9y = 18
6x – 6x – 9y = -6x + 18
-9y = -6x + 18
-9y/-9 = -6x/-9 + 18/-9
y = 6/9x – 2
y = 2/3x – 2
A line perpendicular to this line will have a slope of -3/2.
You may recall from geometry that an inscribed angle (an angle with its vertex on the circle) has a measure that is half the arc it intercepts. Because ∠BAC intercepts the circle at the endpoints of a diameter of the circle, it intercepts an arc of 180°. Therefore ∠BAC is a right angle and the triangle must be a 30-60-90 special right triangle. If you remember the angle relationships for the 30-60-90 triangle you can label the sides of the triangle.
Now find the area of the triangle using the area formula.
A = ½bh
A = ½(6)(6√3)
A = 18√3
You can find the distance between the two cities on the map using the Distance Formula.
If you don’t recall the Distance Formula, you can sketch the problem out and use the Pythagorean Theorem (Distance Formula is really Pythagorean Theorem “in disguise” … for more on this, see here).
a² + b² = c²
5² + 8² = c²
25 + 64 = c²
89 = c²
√89 = c
To find the actual distance between the two cities, multiply your answer by 5 since each unit on the map represents 5 miles.
Distance = 5(√89) ≈ 47
The amplitude of the function is the distance between its horizontal midline and its highest (or lowest) point. You can also think of it as half the distance between its highest and lowest points. For this function, the amplitude is 3.
The period of the function is the horizontal distance required for the graph to complete one full instance of the repeating pattern. The fancy math way of saying this is that it is the value of p such that f(x) = f(x + p) for all values of x. I think I like “how long it takes to complete one full pattern” better! The period for this function is 2.
Therefore, the value of a + p is 3 + 2 = 5
If you have questions about these solutions or anything else to do with the ACT, send us an email at firstname.lastname@example.org