# Tutor profile: Julie L.

## Questions

### Subject: Trigonometry

Prove the following identity by showing justifying each step (you can assume knowledge of the sum and difference formulas for trigonometric functions): sin(2x)=2sin(x)cos(x)

We will start on the left side of the equation and use the following steps to prove the identity: sin(2x) = sin( x + x) by the fact that multiplication is repeated addition = sin(x)cos(x) + cos(x)sin(x) by the sine of the sum of two angles formula = sin(x)cos(x) + sin(x)cos(x) by commutativity of multiplication = 2sin(x)cos(x) by repeated addition being the same as multiplication Thus, we have proven that sin(2x)=2sin(x)cos(x).

### Subject: Calculus

Find an integral expression that represents the area under the curve f(x) = x^2 + 3 on the interval from [0,4]. Do not actually find the area.

First, be sure the given function is positive (above the x-axis) on the given interval. We know that f(x) = x^2 + 3 represents a parabola opening up with a vertex of (0,3). So f(x) is positive on the [0,4] interval. Then we can represent the area under the curve of f(x) by setting up an integral and bounding it by 0 on the bottom, 4 on the top, placing the f(x) function in for the integrand and using dx as the differential. So the following would represent the area: Area = \int_{0}^{4}(x^2+3)dx

### Subject: Algebra

A ball is dropped from the top of a 320 foot tall building. Determine the time, t, in seconds, at which the ball will hit the ground using the position function given by s(t) = -16t^2 + 320 (which is measured in feet).

The position at which the ball will hit the ground is found by determining the time at which the position of the ball is zero feet above the ground ( in other words, when s(t) = 0). So to solve this problem, follow these steps: 1. Set position function equal to zero: -16t^2 + 320 = 0 2. One way to solve the equation would be to isolate the variable. To do this, first subtract 320 from both sides of the equation and then reduce by combining like terms: -16t^2 + 320 - 320 = 0 - 320 -16t^2 = -320 3. To continue to solve, divide both sides by -16 and reduce by reducing fractions: (-16t^2)/(-16) = -320/-16 t^2 = 20 4. To complete the solving steps, take the square root of both sides to eliminate the square and then reduce. Don't forget to put plus and minus on the right side as every positive integer has two distinct square roots! Sqrt(t^2) = +/- Sqrt(20) t = +/- Sqrt(20) 5. Then, since time should be positive, eliminate the negative answer. t = Sqrt(20) 6. Notes on the Final Answer: If you need an exact, but reduced, final answer, factor the 20 as 4*5 and then reduce the square root of 4 to 2 by placing a 2 outside of the square root. There would still be a factor of 5 under the root since 5 is not factorable into perfect squares. So in this case, your answer would look like this (don't forget to put units on the final answer): t = Sqrt(4*5) = 2Sqrt(5) seconds If you need a decimal answer, use a calculator and round appropriately: t = Sqrt(20) is approximately 4.47 seconds if rounding to the nearest hundredth. Let me know if you have questions about this solution!

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