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Etan M.

Student at Purdue University

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Pre-Calculus

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Question:

A specific function has the following equation: $$ \frac{x+3}{x^2 - 16} $$ Of which, $$ x $$-values does the function output no solution?

Etan M.

Answer:

An equation outputs no solution whenever there is a zero (0) in the denominator. So, in order to find out which $$ x $$-values show no solution, we must find out when the denominator equals zero (0). This can be displayed as: $$x^2$$- 16 = 0 Now, simplify the expression: $$x^2$$ = 16 To isolate $$x$$, we must take the square root of both sides: $$x$$ = 4 But since $$(-4)^2$$ also equals 16, $$x$$ also equals -4, so: The function outputs no solution when $$x$$ = 4 and $$x$$ = -4.

Trigonometry

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Question:

One of the smaller sides of a right triangle has a measurement of 5 feet, and the angle that it creates with the longest side is 30 degrees. What is the length of the longest side?

Etan M.

Answer:

We can solve this problem using the basic trigonometric identities. But instead of using any random identity, we must find out which one to use. The sine (sin) of an angle is equal to the side that doesn't touch the angle (opposite) divided by the longest side of the triangle (hypotenuse). We are dealing with the side that DOES touch the angle, so this trigonometric identity won't be useful. The tangent (tan) of an angle is equal to the side that doesn't touch the angle (opposite) divided by the side that touches the angle (adjacent). Since we are trying to solve for the longest side, this trigonometric identity isn't useful either. The cosine (cos) of an angle is equal to the side that touches the angle (adjacent) divided by the longest side of the triangle (hypotenuse). Because we are given the adjacent side length and the angle, we can find the length of the longest side (hypotenuse). The equation for that is represented as: cos(angle) = adjacent / hypotenuse The angle we are given is 30 degrees, and the length of the side touching it (adjacent) is 5 feet. Plug those numbers into the equation: cos(30) = 5 / hypotenuse Now simplify the equation: cos(30) = 5 / hypotenuse cos(30) * hypotenuse = 5 hypotenuse = 5 / cos(30) When using a calculator, make sure the mode is in degrees, or else your number will be completely different. hypotenuse = 5 / cos(30) hypotenuse = 5 / 0.866 hypotenuse ≈ 5.774 The length of the longest side is around 5.774 feet.

Algebra

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Question:

For Thanksgiving, Emily, Joseph, and Sofia all receive different amounts of money based on their age. Joseph was given $4 less than one-half of what Sofia got, and Emily received $6 more than twice of what Joseph got. If Joseph was given $20, how much did Emily and Sofia receive?

Etan M.

Answer:

We already know that Joseph has $20, so we must find the relationship between Joseph and the rest of the kids. First, give each of the kids a specific variable. Emily will be represented by E, Joseph by J, and Sofia by S. We know that Joseph was given $4 less than one-half of what Sofia got. Using our variables, we can say that: J = (1/2)S - 4 or 20 = (1/2)S - 4 <-- Because we know that Joseph was given $20, we substituted J with 20 To solve for S or the amount of money Sofia received, we must isolate S from the rest of the equation: 20 = (1/2)S - 4 24 = (1/2)S 48 = S Sofia was given $48. We also know that Emily received $6 more than twice of what Joseph got. Using our variables, we can say that: E = (2)J + 6 or E = (2)(20) + 6 <-- Because we know that Joseph was given $20, we substituted J with 20 To solve for E or the amount of money Emily received, all you need to do is multiply and add the numbers together: E = (2)(20) + 6 E = 40 + 6 E = 46 Emily was given $46.

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