# Tutor profile: Noah D.

## Questions

### Subject: Trigonometry

Suppose you have a right triangle with a hypotenuse that has length 8 and angle between the hypotenuse and adjacent leg of the triangle of $$\theta=43.2$$ Calculate the length of the adjacent side and the opposite side of the triangle.

For any right triangle we have the following: $$sin(\theta)=\frac{opp}{hyp}$$ $$cos(\theta)=\frac{adj}{hyp}$$ $$tan(\theta)=\frac{opp}{adj}$$ Given these simple relations we can solve for any number of properties of a right triangle. Step one: Calculate the length of the opposite side: $$sin(43.2)=\frac{x}{8}$$ *multiply each side by 8 $$8sin(43.2)=x$$ *plug into calculator $$x=-5.6$$ (we can ignore the negative here since there are no negative lengths!) Step two: Adjacent side $$cos(43.2)=\frac{x}{8}$$ $$8cos(43.2)=x$$ $$x=-5.67$$ (we can again ignore the negative sign So we know the sides of our triangle are 8, 5.6 and 5.67 respectively.

### Subject: Calculus

Find the maximum and minimum of the following function and subsequent interval: $$f(x)=x^{4}-x^{3}$$ on the interval $$ [0,7] $$

Step 1: Take the derivative of $$f(x)$$ $$f'(x)=4x^{3}-3x^{2}=x^{2}(4x-3))$$ Note that $$f'(x)=0$$ when $$x=0$$ and $$x=(\frac{3}{4})$$ These are noted as "critical points". Step 2: Evaluate the function at the zeros and the endpoints of the interval itself (i.e. 0, 7, and $$(\frac{3}{4})$$ $$f(0)=0$$ $$f(7)=7^{4}-7^{3}=2058$$ $$f(\frac{3}{4})=(\frac{3}{4})^{4}-(\frac{3}{4})^{3}=-\frac{27}{256}$$ Thus $$f(x)$$ has a maximum of 2058 at $$x=7$$ and a minimum of $$-\frac{27}{256}$$ at $$x=\frac{3}{4}$$

### Subject: Algebra

Erin and her colleagues are ecologists on an excursion working on modelling the spatial distribution of the Red-Tailed Deer and Cougars in Oregon, USA. Before the excursion Erin decided that she would take data from an area of $$1km^{2}$$. Erin knows from previous studies that for every 100 deer observed in a $$1km^2$$ of this region there is 1 cougar in the habitat. After a 2 days of recording data in $$1km^{2}$$ Erin had observed 224 deer. She also noted that there were $$4km^{2}$$ of habitat in total that she was interested in. Erin wanted to make a quick estimation as to how many cougars she could expect to see in $$4km^{2}$$ but first needed to set up an equation to do so. Using a set of algebraic equations to solve this, how many cougars would Erin have estimated there to be in the $$4km^{2}$$ habitat? (Note: the ecological statistics in this problem are likely not accurate but rather used as a basis for understanding algebraic systems)

Story problems like this one can often be confusing to dissect. What information is important and what information isn't? Sometimes it can helpful to write what we know and what we need to know, for example: What we know: In $$1km^{2}$$ there are 224 deer. We are given a ratio of $$\frac{100 deer}{1 cougar}$$ in $$1km^{2}$$ There is $$4km^{2}$$ of total habitat. What we want to know: How many cougars are there in the $$4km^{2}$$ habitat? We can approach this problem in a variety of ways but the problem itself specifically calls for us to set up a set of algebraic equations to solve the problem. The first step is to figure out what are the variables: We first need to figure out how many cougars there are in $$1km^{2}$$: $$\frac{224deer}{Xcougars}=\frac{100deer}{1cougar}$$ We need to isolate X in order to solve for it. First we multiple each side by $$X$$: $$(X)(\frac{224}{X})=(\frac{100}{1})(X)$$ $$\Rightarrow$$ $$224=100X$$ Now we divide each side by 100 to isolate X: $$\frac{224}{100}=\frac{100X}{100}$$ $$\Rightarrow$$ $$\frac{224}{100}=X$$ $$\Rightarrow$$ $$=2.24$$ Which because we can't have .24 of a cougar, we round down to the nearest integer: 2. Now we know that in $$1km^{2}$$ of habitat there were 2 cougars. The problem wants us to figure out how many cougars there would be in $$4km^{2}$$ of habitat using algebra. Thus we'd use the same method as above: $$\frac{2cougars}{1km^{2}}$$=$$\frac{Ycougars}{4km^{2}}$$ Once again we are trying to isolate the variable, this time Y: $$4(\frac{2}{1km^2})=4(\frac{Y}{4km^{2}})$$ $$\Rightarrow$$ $$8=Y$$ Thus we can say that Erin's estimate of how many cougars she would have recorded in the $$4km^{2}$$ is 8.

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