Suppose a 10 foot ladder leans against a way and the base of the ladder is 5 feet from the wall. How far up the wall it the top of the ladder and what angle does the ladder make with the ground? Assume the ground is flat and perpendicular to the wall.
First, we use Pythagorean theorem to find the how far up the wall the top of the ladder is. The Pythagorean theorem is a^2 + b^2 = c^2, where c is the hypotenuse and a and b are lengths of the legs of the triangle. Therefore, 5^2 + b^2 = 10^2 and b= square root of 75, or approximately 8.7 feet. To get the angle, we can use either sine, cosine or tangent. Let's use the values given in the problem, which are the adjacent side and hypotenuse of the triangle. Therefore, cosine of the angle is 5/10, the angle (using a calculator) or knowing the values of sine and cosine from the unit circle, the angle is 60 degrees
In a recent national poll, 1000 people were randomly selected and asked if they approve of the job the President is doing and 468 said they approve. We want to find a 95% confidence interval of the true population proportion of individuals who approve of the job the President is doing. Can we conclude that less than half of the population approve?
For the confidence interval for a proportion, we need the following information, p^, 1-p^, n and Z critical. For a 95% interval, the Z critical value is 1.96. The sample proportion p^ is 468/1000 = .468 and 1 - p^ = .532. The formula for the interval is p^ +/- Zcritical*(square root(p^*(1-p^)/n)). Therefore the confidence interval is .468 +/- .031 = .437 to .499. Since .500 is not in the interval, we can conclude that less than half of the population approve of the job the President is doing.
Suppose a baseball is hit and it's path is defined by the function f(x) =(-1/4)x^2 + 10x + 3. What is the maximum height of the baseball and what is the horizontal distance when the maximum height is reached?
First, one must recognize that the graph of this function is a parabola that opens downward, since the leading coefficient is negative. If the equation is in the form of ax^2 + bx + c, the x-coordinate of the vertex of the parabola is x = -b/2a. In this problem, a = -1/4 and b = 10. Therefore the x-coordinate of the vertex is x = (-10/2(-1/4)) = 20. This is the horizontal distance the ball traveled when it reached it's maximum height. The height can be obtained by taking f(20), which is (-1/4)(20^2) +10(20) + 3 = 103 feet.