# Tutor profile: Devin F.

## Questions

### Subject: Trigonometry

Aubree is standing at the base of a tall office building in downtown New Orleans. She walks 500 feet away from the base of the office building and looks toward the top of the building. Using a device called a clinometer, she is able to measure her angle of elevation at $$60^o$$ as looks at the very top of the building. How tall is the office building?

Answer: 866 ft. If you were to model this problem visually, you may see that we're basically working with a right triangle scenario. The office building would be a vertical leg of the triangle and the distance Aubree is standing from the base would be a horizontal leg of the triangle, with those two legs, or sides, forming the right angle. If we draw an imaginary line from where Aubree is standing the where she is looking towards the top of the building, that imaginary line would be the hypotenuse. So, we know we have a right triangle scenario. The only other information this problem provides is an angle of elevation and a distance. Whenever a scenario involves a triangle, a length, and an angle, trigonometry should immediately jump out to you! Trigonometry relates angles and sides, which allows us to solve major problems with limited information. When you think trigonometry, you'll probably think of SOH CAH TOA - the acronym that helps us to remember the 3 popular trigonometric functions; sin, cosine, and tangent. In order to solve this problem, we're going to have to determine which trigonometric function we're going to use: sine, cosine, or tangent. First, we need to identify which parts of this office building triangle scenario is important to our problem. Being that Aubree is looking at the building at an angle, Aubree is our reference angle, or we can call that angle $$\theta$$. Since the office building is opposite of where she is looking, the office building would be considered our opposite leg. Since we know that Aubree is standing 500 feet from the base, we know the base is our adjacent leg, which means the leg that is touching $$\theta$$. Since we're trying to find the office building, lets call this side length, or leg $$'x'$$. Now, we have: $$opposite \ leg = x$$ $$adjacent \ leg = 500$$ $$\theta = 60^o$$ Out of the 3 trig functions, only ONE of them involves the triangle parts in our scenario - TANGENT aka TOA! Let's write this TOA function out completely: $$\tan \theta = \frac{opposite \ leg}{adjacent \ leg}$$ If we plug in our known values, we have: $$\tan 60^o = \frac{x}{500}$$. Now, we just have use some equation solving skills from algebra to solve for x. This just so happen to be an equation that we can solve in one step. We're going to undo, or cancel out, this division of 500 by multiplying 500 on both sides of the equation: $$500 * \tan 60^o = \frac{x}{500} * 500$$ $$500 * \tan 60^o = x$$ Now, we'll evaluate $$500 * \tan 60^o $$ to get 866.03 (rounded to nearest hundredth). So, $$866 = x$$ , which is the approximate height of our office building in feet.

### Subject: Geometry

A rectangle is 5 times long as it is wide. What is the perimeter of the rectangle?

Answer: $$P = 10w + 2w $$ Perimeter of a shape is the distance around the outside, or all of the sides added together. One thing we should know about a rectangle is that it has 4 sides - 2 of those sides have the same width and the other 2 sides have the same length. To write this as an equation, $$ P = (length + length) + (width + width) $$ or as variables $$ P = (l + l) + (w + w) $$ and simplified further to $$ P = 2l + 2w $$ (since there are 2 lengths and 2 widths). Now that we have an equation, let use it for our rectangle scenario. A rectangle is 5 times long as it is wide. This means that whatever the width, the wide side, the length is 5 times that width, or $$ 5 * width = 5w $$. We don't know the actual width so we'll just call it $$ 'w'$$. So, we have $$length = 5w$$ and $$width = w.$$ Let's plug it in to our perimeter equation: $$ P = 2l + 2w $$ $$ P = 2(5w) + 2(w) = 10w + 2w $$

### Subject: Algebra

A local gym charges $10 per month for membership plus an upfront $20 annual fee. If 'm' represents # of months, write a function to model the total cost of a gym membership for a member who started in this year.

Answer: $$ f(m) = 10m + 20 $$ Since the cost of our gym membership depends on the number months, 'm' (# of months) will be our dependent variable and $$f(m)$$ will represent the total cost. $$F(m)$$ (function of m) represents total cost because the equation only works, or functions, if we know the number of months. Charging "$10 per month" means to add $10 repeatedly for each month that passes. When you think of repeated addition, you're multiplying! Sooo . . 10 times the # of months = $$10m$$ Since there is an upfront fee of $20, we'll have to add this on to our gym membership cost. We know for each month it will cost $$$10m$$. So, now we have $$ $10m + $20 $$. Altogether, our function shall be: $$ f(m) = 10m + 20 $$

## Contact tutor

needs and Devin will reply soon.