Hopefully you were able to find some time and a quiet place this weekend to try our Five Problems for Your Weekend (5/15/15). Below you’ll find the fully worked-out solutions to those problems.
If a sequence is geometric, you can find the common ratio between the terms by dividing any term by the term that immediate preceeds it. For this sequence, the common ratio is 162/243 = 2/3. Now you can continue the sequence by repeatedly multiplying by 2/3.
Fourth term: 108(2/3) = 72
Fifth term: 72(2/3) = 48
Sixth term: 48(2/3) = 32
This problem can be solved by using the Pythagorean Theorem with the ladder as the hypotenuse (longest side) of the right triangle formed by the ladder, the building and the ground.
a² + b² = c²
x² + 6² = 20²
x² + 36 = 400
x² + 36 – 36 = 400 – 36
x² = 364
x = √364
x ≈ 19.1
In order to write the equation of a circle, you need to know both the coordinates of the center of the circle and its radius. The center of the circle is the midpoint of either of the diagonals of the square. It is located at (6, 4). The length of the side of the square, which is 8 units, is also the diameter of the circle. Therefore, the radius is 4 units.
For a circle with a radius of r units and a center located at the point (h, k) the equation is given by:
(x – h)² + (y – k)² = r²
(x – 6)² + (y – 4)² = 4²
(x – 6)² + (y – 4)² = 16
This is a good problem to use the Picking Numbers strategy. Let’s suppose her old garden had a diameter of 6 feet and now she’s going to double that to 12 feet.
Diameter = 6; radius = 3
Area = πr² = π(3)² = 9π
Circumference = πd = 6π
Diameter = 12; radius = 6
Area = πr² = π(6)² = 36π
Circumference = πd = 12π
Now you can see that the area is four times as large and the circumference is doubled.
Average problems are often really about the total. Let’s organize our information and see what we can figure out.
Total boxes sold by juniors: 20 x 18 = 360
Total boxes sold by seniors : 30 x 24 = 720
Total boxes sold by both classes: 360 + 1080
Total number of students: 20 + 30 = 50
Average number of boxes sold: 1080/50 = 21.6
If you want to read more about Average Problems, go here.
If you have questions about these problems or anything else to do with the ACT, send us an email at firstname.lastname@example.org.