Question

A student tosses an apple off the top of the castle. The path of the apple can be modeled by the equation y = -4x² + 2x + 35 where y is the height of the apple and x is the time in seconds since the toss.

a. How long will it take for the apple to hit the ground? Show all work to receive full credit.

OPTION 1: 2.5 seconds
OPTION 2: 3.5 seconds
OPTION 3: 4 seconds
OPTION 4: 5 seconds

Answer 1

To determine the time it will take for the apple to hit the ground, we solve the quadratic equation y=0 using the quadratic formula. With the coefficients plugged in, we find that it will take approximately 2.72 seconds for the apple to hit the ground, making Option 2 the closest answer.

Step-by-step explanation:

The question asks about the time it will take for the apple to hit the ground when following the path defined by the quadratic equation y = -4x² + 2x + 35, where y is the height and x is the time in seconds. The apple will hit the ground when the height y is equal to zero. To find this time, we need to solve for x in the equation when y is set to zero.

Setting the equation to zero:

0 = -4x² + 2x + 35

This is a quadratic equation in standard form, which can be solved using the quadratic formula x = (-b ± sqrt(b² - 4ac)) / (2a), where a is the coefficient of x², b is the coefficient of x, and c is the constant term. In this equation, a = -4, b = 2, and c = 35.

Plugging these values into the quadratic formula:

x = ( -2 ± sqrt(2² - 4(-4)(35))) / (2(-4))

x = ( -2 ± sqrt(4 + 560)) / (-8)

x = ( -2 ± sqrt(564)) / (-8)

x = ( -2 ± 23.76) / (-8)

Since time cannot be negative, we discard the solution with -2 - 23.76 and only consider x = ( -2 + 23.76) / (-8), which yields:

x ≈ 2.72 seconds

As 2.72 seconds is closest to Option 2: 2.5 seconds, this is the approximate time it will take for the apple to hit the ground.