# Tutor profile: Tj F.

## Questions

### Subject: Trigonometry

Peter needs a friend to help him measure Peter's height, so he decides to lean his head against the wall and think of people he can ask. Peter is leaning to where there is a 60-degree angle between himself and the floor and his head is touching the 5-foot mark on the wall. Using this information, be a great friend and find Peter's height.

In order to find Peter's height, we should draw a graph. When we draw the graph, the situation seems to model that of a right triangle where Peter is the hypotenuse of the right triangle. Fortunately, we can use trigonometry to figure out Peter's height which corresponds to the length of the hypotenuse To find Peter's height in this scenario, we need to use our given information including the 60-degree angle and the fact that Peter touches the wall at the 5-foot mark. Remember: $$sin(\theta) = \frac{opposite}{hypotenuse}$$ We can use this formula for the given situation where theta is 30 degrees, the opposite side to theta is 5, and the hypotenuse is Peter's height in feet. Plugging in our given values into the necessary place, $$ sin(60{\circ}) = \frac{5}{hypotenuse} $$ Solving this equation, $$ hypotenuse = \frac{5}{sin(60{\circ})}$$ $$ hypotenuse = \frac{10}{\sqrt3}$$ = $$ 5.7735 $$ feet Therefore, Peter measures approximately 5.7735 feet.

### Subject: Calculus

Find the area between the two curves, $$ y = x^2 $$ and $$ y = -x^4 + 5 $$ bounded by numerical values, -1 and 1.

In order to find the area $$ A $$ between two curves, we must create a definite integral of the difference between the two functions that the area is bounded by. The formula to create this integral is: $$ A = \int_a^b \mathrm{[f(x) - g(x)]}\mathrm{d}x $$ Keep in mind that $$ f(x) $$ is the function that serves as the upper bound for the area we are trying to find and $$ g(x) $$ is the function that serves as the lower bound for the area we are trying to find. $$ a $$ and $$ b $$ are the numerical values that we are evaluating this integral between. According to the graphs of both functions, $$ f(x) = -x^4 + 5 $$ and $$ g(x) = x^2 $$ Plugging into the above formula to find our area, $$ A = \int^1_{-1} \mathrm{[(-x^4 + 5) - (x^2)]}\mathrm{d}x $$ Simplifying the integral, $$ A = \int^1_{-1} \mathrm{(-x^4 + 5 - x^2)}\mathrm{d}x $$ Now, we must integrate the expression $$ A = \mathrm{[-\frac{1}{5}x^5 + 5x - \frac{1}{3}x^3]}^1_{-1} $$ Plugging in the numerical bounds we are integrating over, $$ A = \mathrm{[(-\frac{1}{5}(1)^5 + 5(1) - \frac{1}{3}(1)^3)-(-\frac{1}{5}(-1)^5 + 5(-1) - \frac{1}{3}(-1)^3)]}$$ Evaluating this numerical expression, $$ A = \mathrm{[(-\frac{1}{5} + 5 - \frac{1}{3})-(-\frac{1}{5}(-1) + (-5) - \frac{1}{3}(-1))]}$$ $$= \mathrm{[(-\frac{1}{5} + 5 - \frac{1}{3})-(\frac{1}{5} - 5 + \frac{1}{3})]}$$ $$= \mathrm{(-\frac{1}{5} + 5 - \frac{1}{3} - \frac{1}{5} + 5 - \frac{1}{3})}$$ $$= \mathrm{(-\frac{3}{15} + \frac{75}{15} - \frac{5}{15} - \frac{3}{15} + \frac{75}{15} - \frac{5}{15})}$$ $$= \mathrm{(-\frac{6}{15} + \frac{150}{15} - \frac{10}{15})}$$ $$= \mathrm{\frac{134}{15}}$$ Thus, the area bounded by the curves $$ y = x^2 $$ and $$ y = -x^4 + 5 $$ between the numerical bounds of -1 and 1 is $$\mathrm{\frac{134}{15}}$$

### Subject: Algebra

Zeke and Amari both like to sell hats, and they have sold 14 hats in the past hour altogether. Zeke's hats are $6 each and Amari's hats are $5 each. If they have made $74 combined in the past hour, how many hats have Zeke and Amari each sold in the past hour?

In order to find a solution to this problem, you must set up a system of equations. First, represent the number of hats Zeke sold in the past hour as $$ z $$ and the number of hats Amari sold in the last hour as $$ a $$. The system would appear as follows: \begin{array}{rcl} {z + a = 14} \\ {6z + 5a = 74}\\ \end{array}. (1) First, we will use substitution in order to find both $$ a $$ and $$ z $$. We will take the first equation, $$ {z + a = 14} $$, and solve for $$ a $$. This gives us the equation $$ {a = 14 - z} $$ (2) Second, we will plug in $$ {14 - z} $$ for $$ a $$ in the second equation. This makes the second equation: $$ {6z + 5(14 - z) = 74} $$ After simplifying, $$ {6z + 70 - 5z = 74} $$ $$ 1) $$ Distribute $$ 5(14 - z) $$ $$ {z + 70 = 74} $$ $$ 2) $$ Subtract $$ 6z - 5z $$ $$ {z = 4} $$ $$ 3) $$ Subtract $$ 70 $$ from both sides of the equation We have found that Zeke sold 4 hats in the past hour, but we still need to find the number of hats Amari has sold in the past hour. (3) Third, we will plug 4 in for $$ z $$ back into the first equation and solve for $$ a $$. Looking at our first equation $$ {z + a = 14} $$ We will now plugin 4 for $$ z $$ $$ {4 + a = 14} $$ Subtracting 4 from each side of the equation, $$ {a = 10} $$ (4) The solution to this problem is that Zeke has sold 4 hats in the past hour and Amari has sold 10 hats in the past hour. To check the solution, you can plug in 4 for $$ z $$ and 10 for $$ a $$ and the equations should come out true as shown below. $$ {z + a = 14} $$ $$ {4 + 10 = 14} $$ $$ {14 = 14} $$ $$ {6z + 5a = 74} $$ $$ {6(4) + 5(10) = 74} $$ $$ {24 + 50 = 74} $$ $$ {74 = 74} $$ This shows that our solution is correct.

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