Question

2. The function H(t)=−16t 2

+88t gives the height (in feet) of golf ball after t seconds. Round answers to two decimal places as needed. All answers must include appropriate units of measure. a) Determine the maximum height of the golf ball. Show your work. Write your answer in a complete sentence. b) How long does it take for the ball to hit the ground? Show your work. Write your answer in a complete sentence. c) Identify the Vertical Intercept ( y-int). Write it as an ordered pair and interpret its meaning in a complete sentence. d) Determine the practical domain of H(t). Use interval notation and include units. e) Determine the practical range of H(t). Use interval notation and include units.

Answer 1

a) The maximum height of the golf ball is 66 feet. b) It takes 5.5 seconds for the ball to hit the ground. c) The vertical intercept is (0, 0), indicating the ball starts at a height of 0 feet. d) The practical domain is [0, ∞) seconds. e) The practical range is (-∞, 66] feet.

Step-by-step explanation:

a) To determine the maximum height of the golf ball, we need to find the vertex of the quadratic function H(t) = -16t^2 + 88t. The formula to find the x-coordinate of the vertex is given by x = -b/2a, where a = -16 and b = 88. Plugging in these values, we have t = -88/(2*(-16)) = 2.75 seconds. To find the y-coordinate of the vertex, we can substitute t = 2.75 into the function H(t). H(2.75) = -16(2.75)^2 + 88(2.75) = 66 feet. Hence, the maximum height of the golf ball is 66 feet.

b) The ball hits the ground when H(t) = 0. To find the time it takes, we can solve the quadratic equation -16t^2 + 88t = 0. Factoring out -16t, we get -16t(t - 5.5) = 0. So, t = 0 or t = 5.5. Since we are only interested in the positive root, the ball takes 5.5 seconds to hit the ground.

c) The vertical intercept (y-int) represents the initial height of the golf ball when t = 0. In this case, the vertical intercept is given by the function evaluated at t = 0. H(0) = -16(0)^2 + 88(0) = 0. Therefore, the vertical intercept is (0, 0), meaning the ball starts at a height of 0 feet.

d) The practical domain of H(t) represents the values of t for which the function is defined. In this case, since the function represents the height of a golf ball, the practical domain would be all non-negative values of t. Using interval notation, we can express this as [0, ∞). The units for the domain would be seconds.

e) The practical range of H(t) represents the possible heights of the golf ball. Since the function is a quadratic, its range will depend on whether the coefficient of the quadratic term (a) is positive or negative. In this case, since a = -16 is negative, the graph opens downwards, indicating that the maximum height is limited. The range would be all values less than or equal to the maximum height, which we found to be 66 feet. Using interval notation, the practical range would be (-∞, 66]. The units for the range would be feet.