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Trigonometry
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Question:

Suppose that $$x$$ and $$y$$ are numbers such that $$\sin(x+y) = 0.3$$ and $$\sin(x-y) = 0.5$$. Then $$\sin (x)\cdot \cos (y) =$$

Spencer P.

Recalling our trig identities, we know that $$\sin(x+y) = \sin x \cos y + \sin y \cos x$$ and $$\sin(x-y) = \sin x \cos y - \sin y \cos x$$. Making these substitutions to our original equations, they become $$(1)$$ $$\sin{x}\cos{y}+\sin{y}\cos{x}=0.3$$ and $$(2)$$ $$\sin{x}\cos{y}-\sin{y}\cos{x}=0.5$$ Adding these two equations together, we get $$2 \sin x \cos y = 0.8$$, which simplifies to $$\sin x \cos y = 0.4$$

Geometry
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Question:

Increasing the radius of a cylinder by 6 units increased the volume by $$y$$ cubic units. Increasing the height of the cylinder by 6 units also increases the volume by $$y$$ cubic units. If the original height is 2, then what is the original radius?

Spencer P.

For a cylinder with radius $$r$$ and height $$h$$, the volume of the cylinder is $$V = \pi r^2h$$. Since we know that the original height is 2, the original volume of the cylinder is $$2\pi r^2$$. When we increase the radius by 6 units, that cylinder's volume will be expressed as $$2\pi (r+6)^2$$, and this area is $$y$$ units larger than our first volume. We can write this all in one neat equation: $(2\pi (r+6)^2 - 2\pi r^2 = y$) When we expand and simplify this equation, it can be expressed as $$24\pi r + 72\pi = y$$. Since we have two variables in our equation, $$r$$ and $$y$$, we will actually need a second equation with those two variables in order to solve this problem. Let's consider what happens to the volume of the cylinder when we change the height. Our original height is 2, and when we increase the height by 6 units (for a total height of 8 units), our volume increases by $$y$$ units. This can be expressed in one equation as $(8\pi r^2 - 2\pi r^2 = y$) which simplifies to $$6\pi r^2 = y$$. Now, we can set these two equations together and solve for our radius $$r$$. $(24\pi r + 72\pi = 6\pi r^2$) which reduces to $(r^2 - 4r - 12 = 0$) Here we have a simple quadratic equation that we can factor: $((r-6)(r+2)=0$) These factors tell us that $$r = 6$$ or $$r = -2$$. Since our radius cannot be negative, our original radius must be 6 units.

Algebra
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Question:

I have written down one integer two times and another integer three times. The sum of my five numbers is 100, and one of the integers is 28. What is the other number?

Spencer P.

Since my two numbers are integers, it is important to remember that an integer is a number that can be written without a fractional component. Numbers like 8, 0, and -37 are all integers. For now, I'm going to call my first integer $$x$$ and my second integer $$y$$. I wrote down the first integer twice and the second integer three times. When I add those five numbers together, I get this equation: $(x + x + y + y + y = 100$) which we can rewrite as $(2x + 3y = 100$) I also told you that one of the numbers is 28. Now, we have to figure out whether 28 is my first or second number. If 28 is my first number, then $$x=28$$. We can plug 28 in for $$x$$ and solve the equation above for $$y$$. When we do that, we get $$y = \frac{44}{3}$$, which is not an integer. This means that 28 is my second number, or $$y = 28$$. Now we can plug in 28 for $$y$$ and find our first integer. If you do the work correctly, you should get $$x = 8$$. Thus, my other number is 8.

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