A ladder leans up against a building and forms a 32 degree angle with the ground. The ladder reaches a height of 40 feet vertically up the building. How tall is the ladder?

We have an angle and two sides of a triangle. The trig function we use depends on which sides we have. The ladder is the hypotenuse of the triangle and the vertical height is opposite the 32 degree angle. Since we have opposite and want the hypotenuse, we want to use sine (SOHCAHTOA) and set up a trig ratio. We have sin32 = \frac{opp}{hyp}. sin32 = \frac{40}{hyp}. Next, find sin32 on the calculator (make sure it's in degree mode!) 0.53 = \frac{40}{hyp}. We multiply both sides by the hyp and divide by 0.53. \frac{40}{0.53}= 75.47. Therefore, the ladder is 75.47 feet.

A random sample of average course grades of 100 AP students from a particular high school was taken and it was determined that the sample mean is an 87 with a standard deviation of 8. Assuming the conditions for inference are met, create and interpret a 95% confidence interval for the mean course grade for all students at this school.

We need to create a 95% confidence interval for the population mean, using the formula \bar{x} \pm z* \frac{s}{\sqrt{n}}. Next, we substitute the mean of 87, standard deviation of 8, and sample size of 100. 87 \pm z* \frac{8}{\sqrt{100}}. Z* is found either using a table or a calculator. I tell my students to use the command InvNorm on the graphing calculator with an area of 97.5%. InvNorm (.975, 0, 1 ) gives us 1.96. The reason we use 97.5% is because the 95% area is in the middle of the curve. That leaves 5% area left, 2.5% on each end. Adding the 2.5% to the 95% gives us 97.5%. On the table, look for .975 and follow over to the z value of 1.96. Substituting this value for z* gives us 87 \pm 1.96 \frac{8}{\sqrt{100}}. Calculating the value for the margin of error gives us 87 \pm 1.568, Adding and subtracting from 87 results in the interval (85.432, 88.568). AP's interpretation is "we are 95% confident that the interval (85.432, 88.568) captures the true population mean course grade for all AP statistics students at this school." Make sure to state the word "mean" and include the context of the problem.

Find the time for a ball to hit the ground given the quadratic equation h = 3 + 14t - 5t^{2}, where h is height in feet and t is the time in seconds.

In order to find the time for the ball to hit the ground, we need the height to be 0 feet. So we set the quadratic equation = 0 and put it in standard form. 5t^{2} - 14t - 3 = 0. One of the easiest ways to solve quadratics is by factoring. This means we need to create two parentheses that multiply to get the polynomial above. This is done in the following steps: 1. Since the leading coefficient is a 5, multiply 5 X -3 to get -15. 2. Find factors of -15 that add to -14 (1 and -15). 3. Rewrite the quadratic with 4 terms (5t^{2} + 1t - 15t - 3) 4. Split the 4 terms into groups of 2 and factor out the GCF on each side. t(5t+1) - 3(5t+1) the parentheses should be the same. 5. Combine the "outsides" into one parenthese (t - 3)(5t + 1). 6. Set each parenthese = 0 and solve. t = 3, t = -1/5. 7. Since time can't be negative, our answer is t = 3 seconds.