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Gabriella N.
Third Year University Student
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Trigonometry
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Question:

A $$12$$ft plank connects a point on the ground to the bed of a truck. If the plank makes a $$30$$° angle with the ground, how high is the bed of the truck off the ground?

Gabriella N.

If we think about this in terms of a trigonometric triangle, we know that we have a right triangle where the hypotenuse is $$12$$, the angle is $$30$$°, and we are trying to find the height $$(x)$$. We will be using sine to solve for $$x$$ since we know (based on SOHCAHTOA) that the height $$(x)$$ is opposite of the $$30$$° angle. To set up our equation we have $$\sin(30°) = \frac{x}{12}$$ Based on the Unit Circle we know that $$\sin(30°) =\frac12$$ So... $$\frac12 = \frac{x}{12}$$ after cross multiplying to get $$2x = 12$$ We solve for $$x$$ to find the height of the truck bed to be $$6$$ft off the ground.

Geometry
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Question:

Find the surface area (in inches) of a cylinder where the radius is 4in ($$r = 4$$) and the height is 12in ($$h = 12$$)

Gabriella N.

Recall that the formula for the surface area (SA) of a cylinder is $$SA = 2\pi rh + 2\pi r^2$$ Since we know that $$r = 4$$ and $$h = 12$$, we can plug those values into our equation to get $$SA = 2\pi (4)(12) + 2\pi (4^2)$$ Next, we simplify our answer and end up with $$SA = 96\pi + 32\pi = 128\pi = 402.12$$ Our final answer is that the surface area of the cylinder is approximately $$402.12$$ inches

Algebra
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Question:

Multiply the Polynomials $$y=(x^2+4)(2x^2+6)$$

Gabriella N.

Think about this polynomial as (a+b)(c+d) The first step to solve this problem is to multiply a and c $$x^2(2x^2) = 2x^4$$ You take this product and place it after the equal sign $$y = 2x^4$$ Next you multiply a(d) and add that product to the equation above $$x^2(6) = 6x^2$$ so now you have $$y = 2x^4 + 6x^2$$ In the same manner you multiply b(c) and then b(d) and add those products as well $$b(c) = 4(2x^2) = 8x^2$$ $$b(d) = 4(6) = 24$$ so you have $$y = 2x^4 + 6x^2 + 8x^2 +24$$ Finally, you combine the like numbers and end up with $$y = 2x^4 + 14x^2 +24$$

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