(1)
Find the value of x that satisfies
A) 11
B) 7
C) 44
D) 0
The correct answer is A
Explanation
In a radical or absolute-value problem, you first need to isolate the radical or absolute value
Therefore, you have to subtract 1 from both sides of the given equation before doing anything else; doing so gives you this equation
Now you can square both sides to eliminate the radical: 4x
root(4x -8)]^2 = [6]^2]
4x -8 = 36
4x = 44
x = 11
(2)
= If a - b = 8 and ab = 10, then a^2 + b^2
A) 100
B) 164
C) 84
D) 80
The correct answer is C
Explanation
a – b)^2 = a^2 – 2ab + b^2)
You know that (a – b) = 8, so (a – b)^2 = a^2 – 2ab + b^2 = 64
This question asks you for a^2 + b^2, which is (a^2– 2ab + b^2) + 2ab, or 64 + 2(10) = 84
(3)
.The ratio of a rectangle's width to its length is 2:5. If its perimeter is 84 feet, find its width, in feet
A) 10
B) 28
C) 14
D) 12
The correct answer is D
Explanation
When a question tells you the ratio of unknown quantities, you can simply tack an x onto the end of each number
Thus, the length of the rectangle is 5x and the width is 2x. Now you may want to draw and label a rectangle to help you solve this problem
Don't fall into the trap of writing 2x + 5x = 84; a rectangle has four sides, not two
Write either 2x + 5x + 2x + 5x = 84 or 2(2x + 5x) = 84
Either way, the left side is 14x, so 14x = 84, and x = 6
Another trap lurks here: 6 isn't the answer. You represented the width with 2x in your original equation
so the width is 2(6) = 12 feet
(4)
The pyramid has a square base of length 10 cm and a height of 12 cm. Determine the total surface area of all five faces, in square centimeters
A) 180
B) 360
C) 120
D) 320
The correct answer is B
Explanation
The total surface area is the sum of the area of the square and the area of the four triangles. The area of the square is easy to find: It's simply 10 × 10 = 100
The triangles are a little tougher. They don't have a height of 12 because 12 is the height of the pyramid, but the triangles are slanted
However, if you look at the following diagram, you see that you can find the height of the slanted triangles by using the Pythagorean Theorem
The little triangle in the diagram is a right triangle. One leg is 12 (the height of the pyramid). The second leg is half the width of the square, or 5
This triangle is actually the world's second-most famous right triangle, the 5-12-13 triangle.
(If you didn't remember this little tidbit, you could've figured it out with the Pythagorean Theorem)
The hypotenuse, 13, is the altitude of each of the tilted triangles that make up the sides of the pyramid
Because the triangle's area is 1?2 × base × height, each triangle's area is 1?2 × 10 × 13 = 65
The four triangles together have an area of 4 × 65 = 260
Adding in the 100 from the base gives you 360, which is the answer