# Tutor profile: Emily H.

## Questions

### Subject: Pre-Calculus

A quarterback tosses a football to a receiver 40 yards downfield. The height of the football, f(x), in feet, can be modeled by f(x) = -0.025x$^{2}$ + x + 6, where x is the ball's horizontal distance, in yards, from the quarterback. How far from the quarterback does the ball reach maximum height? What is the ball's maximum height?

There are a few ways that we can solve this problem. We do have a quadratic function, and it is asking us about the maximum, which is the vertex of the parabola. The first question asks us for the horizontal distance, which is x. One way to solve this is to use the equation x = -b/2a. A quadratic function is in the form f(x) = ax$^{2}$ + bx + c. Using x = -b/2a with our quadratic function gives us: x = -1 / 2(-.025) = -1/(-.05) = 20 So the ball will reach it's maximum height when it is 20 yards from the quarterback. Now to find the actual maximum height, we need to solve for f(x). We can use x = 20 (the answer we just found). We can plug 20 in for x in our original quadratic function. This gives us: f(x) = -0.025(20)$^{2}$ + 20 + 6 = -0.025(400) + 20 + 6 = -10 + 20 + 6 = 16 So the ball's maximum height is 16 feet.

### Subject: Statistics

A study was conducted on the average height of white oak trees. A random sample of 45 fully-grown white oak trees was taken, and the average height of the sample was found to be 90 feet with a standard deviation of 3.5 feet. Construct a 99% confidence interval for the mean height of all fully-grown white oak trees, and write an interpretation in context.

Before calculating a confidence interval, we should verify that the Central Limit Theorem applies to this problem. We must have a random sample (we do!) and our sample size must be greater than or equal to 25 (ours is 45!). Since the conditions are met, we know that our sampling distribution will be approximately normal, and we can find a confidence interval. Our next step is to find a t-score. Our degrees of freedom, df, will be 45-1 = 44. Using technology, we can find that the best t-score to use is 2.692. Now we can find our margin of error. Our formula is m = t * (s/$\sqrt{n}$) = 2.692 * (3.5/$\sqrt{45}$). This gives us a margin of error of about 1.405. To get the confidence interval, we can add and subtract our margin of error from our sample mean. 90 - 1.405 = 88.595 90 + 1.405 = 91.405 Our confidence interval is (88.595 feet, 91.405 feet). Interpretation: We can be 99% confident that the mean height of all fully-grown white oak trees is between 88.595 feet and 91.405 feet.

### Subject: Algebra

Admission to a children's museum costs $5.50 more for an adult than for a child. A family consisting of 2 adults and 5 children paid $39. Find the cost of admission for each adult and each child.

Let x = cost of admission for each adult, and let y = cost of admission for each child. The first sentence tells us that the price of admission for each adult is $5.50 more than for a child. This gives us an equation: x = 5.50 + y. The second sentence tells us the cost of admission for a family. This gives us another equation: 2x + 5y = 39. The first equation is already solved for x, so we can take what x equals (5.50 + y) and plug it into the second equation in place of x. This gives us: 2(5.50 + y) + 5y = 39 Now we can solve that equation for y! The first thing we should do is distribute the 2: 11 + 2y + 5y = 39 Now we should combine our y's: 11 + 7y = 39 We can move the 11 to the other side of the equation by subtracting it from both sides. This will eliminate it from the left side of the equation: 7y = 28 Now our last step is to divide both sides by 7. y = 4 So the price of admission for a child is $4. We can use this in the first equation to find x: x = 5.50 + y = 5.50 + 4 = 9.50. The price of admission for an adult is $9.50.

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