What is the ycoordinate of the point in the standard (x,y) coordinate plane at which the 2 lines y = ^{x}⁄_{2} + 3 and y = 3x − 2 intersect?

Solution
To find the ycoordinate where the 2 lines y = ^{x}⁄_{2} + 3 and y = 3x−2 intersect, you could set ^{x}⁄_{2} + 3 equal to 3x − 2 because they are both already solved for y. Where they would intersect, their ycoordinates would be equal. To solve ^{x}⁄_{2} +3 = 3x−2, you could add 2 and subtract ^{x}⁄_{2} to both sides to get \(\frac{5x}{2}\) = 5, then multiply by ^{2}⁄_{5} (the reciprocal of ^{5}⁄_{2}) to get x = 2. Then simply substitute 2 for x into either of the initial equations to get y = 4.
When the choir is arranged in rows of 5 people each, the last row is one person short. When the choir is arranged in rows of 6 people each, the last row is still one person short. What is the least possible number of people in the choir?

Solution
The number of people in the choir is 1 short from being able to be divided evenly by both 5 and 6. To find the least possible number of people in the choir, take one less than the lowest number for which 5 and 6 are both factors. The lowest number for which 5 and 6 are both factors is 30. Thus, the least possible number of people in the choir is 30 − 1 = 29.
The hypotenuse of the right triangle ABC shown below is 17 feet long. The cosine of angle C is ^{3}⁄_{5}. How many feet long is the segment AC?

Solution
To find the length of the segment AC in ΔABC, where the length of the hypotenuse is 17, and the cosine of ∠C is ^{3}⁄_{2}, use the definition of cosine: the ratio of the lengths of the adjacent side to the length of the hypotenuse. In ABC cosine of ∠C is the ratio of the segment AC to the length of the hypotenuse.After substituting the length of the hypotenuse, we get ^{3}⁄_{5} = \(\frac{AC}{17}\), and AC = \(\frac{(17 \times 3)}{5}\),or 10.2 feet.
After excavating a lot, workers removed an estimated 7,000 cubic yards of dirt from the area. If this dirt were spread in an even layer over an empty lot with dimensions 30 yards by 64 yards, about how deep, in yards, would the layer of dirt be?

Solution
To find the uniform depth, you would substitute in the formula for volume, V, of a rectangular prism with the height h, length l, and width w, which is V = lwh.
After substituting you should have 7,000 = 30(64)(h), or 7,000 = 1,920h. Thus h = \(\frac{7,000}{1,920}\), or about 3.65, which is between 3 and 4.
For the right triangle ΔABC shown below, what is sin C?

Solution
The sine (sin) of an acute angle in a right triangle is equivalent to the length of the side opposite the angle over the length of the hypotenuse \(\left ( \frac{opp}{hyp} \right )\). To find sin C in ABC, take the length of the opposite side over the length of the hypotenuse, or ^{a}⁄_{b}.
Answer choice G is the tangent \(\left ( \frac{opp}{adj} \right )\), H is the cosecant , J is the cosine \(\left ( \frac{hyp}{opp} \right )\), and K is the cotangent \(\left ( \frac{adj}{hyp} \right )\). If you did not get the correct answer, it would be wise to review
What is the area, in coordinate units, of the triangle in the figure below?

Solution
To solve this problem, remember that the area of a triangle is calculated using the formula A = 1/2(bh), where b is the base of the triangle and h is the height of the triangle. The base of the triangle extends from the origin in the (x, y) coordinate plane, (0,0) to the point (5,0). This means that the base is 5. The height of the triangle extends from the origin in the (x, y) coordinate plane, (0,0) to (0,4). The height of the triangle is 4. Substitute these values into the formula and solve:
A = 1/2(bh)
A = 1/2(5 × 4)
A = 1/2(20)
A = 10
A shoe store charges $39 for a certain type of sneaker. This price is 30% more than the amount it costs the shoe store to buy one pair of these sneakers. At an endoftheyear sale, sales associates can purchase any remaining sneakers at 20% off the shoe store’s cost. How much would it cost an employee to purchase a pair of sneakers of this type during the sale (excluding sales tax)?

Solution
To solve this problem, recognize that the $39 price of the sneakers is 30 percent more than the amount it costs the store to purchase one pair of the sneakers. This can be represented as 130%, or 1.3. Thus, the price that the store pays for the sneakers is $39/1.3, or $30. At the end of the year sales associates get 20% off of this $30 price, therefore paying 80% of the price the shoe store pays. The cost to the employees is $30 × 0.80, or $24.
If the volume of a cube is 64, what is the shortest distance from the center of the cube to the base of the cube?

Solution
To solve this problem, remember that the volume of a cube is equal to (length)(width)(height) or simply (side)3, since all sides of a cube are equivalent in length. To find the length of one side, find the cube root of 64, which is 4 (43 = 64). Because all sides of a cube are equal, the shortest distance from the center of the cube to the base of the cube will equal the midpoint of the length of the cube, which is 4/2, or 2.
What is the slopeintercept form of 9x + 3y − 6 = 0?

Solution
To find the slopeintercept form of the equation 9x + 3y − 6 = 0, you could first add 6 and subtract 9x from both sides of the equation to get 3y = −9x + 6. Then, multiply both sides by ^{1}⁄_{3} to get y = −3x + 2.
If n^{x} · n^{8} = n^{24} and (n^{6})^{y} = n^{18}, what is the value of x + y?

Solution
To solve this problem, remember that, when multiplying the same base number raised to any power, add the exponents. Thus, if n^{x} · n^{8} = n^{24}, x +8 = 24, and x equals 16.Also, remember that, when raising an exponential expression to a power, multiply the exponent and power. So in (n^{6})^{y} = n^{18}, 6y = 18, and y = 3. Therefore, x + y = 16 + 3, or 19.