Clarissa K.

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Trigonometry

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Question:

A hot air balloon is 200 m above the ground. An observer stands on the ground and looks up at the hot air balloon. The distance between the observer and the hot air balloon is 400 m. Determine the angle of elevation.

Clarissa K.

Answer:

In order to solve this problem, we want to imagine the above situation in the form of a right triangle. We are given that the distance between the ground and hot air balloon is 200 m, and this makes up the vertical leg of the triangle. Second, we are given that the distance between the observer on the ground and the hot air balloon in the air is 400 m, which is the hypotenuse. We are trying to solve for the angle of elevation, which is the angle formed between the ground and the line of sight of the observer. So, if we continue to imagine this triangle, we know that from the point of view of the angle of elevation, we are given the hypotenuse and the side opposite to the angle. Therefore, we have to use sine in order to solve for the angle of elevation. Remember that the formula for sine is $$\sin(\theta)=\frac{opposite}{hypotenuse}$$ So, plugging in our given numbers, we get: $$\sin(\theta)=\frac{200 m}{400 m}$$ Solving for θ, we get: $$\sin(\theta)={0.5}$$ $$\theta=\sin^{-1}0.5$$ $$\theta=30^\circ$$

Calculus

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Question:

Find the derivative of the following: $$y=\frac{x^{4}}{3}+e^{2x}+5\pi\ dx$$

Clarissa K.

Answer:

In order to take the derivative of the above equation, we proceed to take the derivative of each individual term. Let's start with $$\frac{x^{4}}{3}$$ To take the derivative of the above, we use the product rule. It is easier to see if we rewrite it as: $$\frac{1}{3}x^{4}$$ So, we take the derivative of $$\frac{1}{3}$$ and multiply it by $$x^4$$. However, the derivative of any constant is $$0$$, so this term cancels out. Now we take the derivative of $$x^4$$, which is $$4x^3$$, and multiply it by $$\frac{1}{3}$$ giving us $$\frac{4}{3}x^3$$. We have found the derivative of the first term. Now we move onto $$e^{2x}$$. To find the derivative of this term, we use the chain rule. We take the derivative of the outer function, which is just $$e^{2x}$$. Now we take the derivative of the inner function, the exponent $${2x}$$, which is just $$2$$. Thus, the derivative of $$e^{2x}$$ is $$2e^{2x}$$. Lastly, we take the derivative of $$5\pi$$. This is a constant, thus its derivative is just $$0$$. So, putting everything we have together, we get that the derivative is: $$\frac{dy}{dx}=\frac{4x^{3}}{3}+2e^{2x}$$

Algebra

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Question:

Solve the following equation for x: $$5(5x+3)-7(x-8)=2x+7$$

Clarissa K.

Answer:

To begin solving this equation, we want to remove parentheses by distribution. So, we distribute the terms that are outside of each set of parentheses to each term within the parentheses. Remember to distribute any negatives that may exist! This will look as follows: $$25x+15-7x+56=2x+7$$ The next step is to combine any like terms. We see that on the left-hand-side, there are two terms that contain an "x" variable, and two terms that don't. So, we can combine the like terms to simplify the left-hand expression as follows: $$18x+71=2x+7$$ The right hand side is already simplified completely, so we leave it alone. Now we wish to get all terms that contain an "x" to one side, and all other terms to the other side. To do this, we add/subtract terms from both sides. Let's begin by subtracting "2x" from both the left and right hand side: $$16x+71=7$$ Now we want to get all other terms to the other side, so we subtract 71 from both sides. This leaves us with: $$16x=-64$$ Our last step is to get the x all by itself, so to do this, we have to "undo" its multiplication with the 16 by division. So, we divide both sides by 16 and get: $$x=-4$$ And that is our final answer!

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