Question

PLEASE HELP!! Brainliest to first answer

Part A: An archer shoots an arrow into the sky where the motion of the arrow can be modeled by the equation() = −5^2 + 20 + 4. Where t is time in seconds and f(t) is height in feet. Find how long it will take the arrow to hit the ground using an algebraic method. Round to the nearest tenth if necessary. Show all of your reasoning used with this algebraic method.

Part B: Using the information in Part A, find how high the arrow will go at its maximum height, and how long it takes for this to occur using an algebraic method. Show all of your reasoning used with this algebraic method. Round to the nearest tenth if necessary.

Answer 1

Answer:

The arrow will take 4.19 seconds to hit the ground

The arrow will go 24 feet to reach its maximum height and it takes 2 seconds to reach this height

Step-by-step explanation:

The quadratic function is f(t) = -5t² + 20t + 4, where

f(t) is the height in feet

t is the time in seconds

If the arrow hit the ground that means h(t) = 0

→ Substitute h(t) by 0 and use the quadratic formula to find t

∵ h(t) = 0

∴ 0 = -5t² + 20t + 4

→ Switch the two sides

∴ -5t² + 20t + 4 = 0

→ Multiply all terms by -1 to make the coefficient of t² positive

∴ 5t² - 20t - 4 = 0

→ The quadratic formula is $t=\frac{-b+-\sqrt{b^{2}-4ac}}{2a}$ , where

• a is the coefficient of t²

• b is the coefficient of t

• c is the numerical term

∵ a = 5, b = -20, c = -4

∴ $t=\frac{-5+-\sqrt{(-20)^{2}-4(5)(-4)}}{2(5)}$

∴ $t=\frac{-5+-\sqrt{400+80}}{10}$

→ Find the values of t in decimal

∴ t = 4.19 seconds OR t = -0.19 ⇒ refused because it is negative

∴ The arrow will take 4.19 seconds to hit the ground

The graph of this quadratic function is a parabola with maximum vertex (h, k), where

• h = $\frac{-b}{2a}$

• k = f(h)

To find the maximum height and the time taking for this to occur find h and k where k is the maximum height and h is the time for this to occur

∵ f(t) = -5t² + 20t + 4

∵ a = -5 and b = 20

∴ h = $-\frac{20}{2(-5)}=-\frac{20}{-10}=2$

∴ The time for the maximum height is 2 seconds

→ Now use it to find k

∵ k = f(h)

∴ k = f(2)

∵ f(2) = -5(2)²+ 20(2) + 4

∴ f(2) = -5(4) + 40 + 4

∴ f(2) = -20 + 40 + 4

∴ f(2) = 24

∴ k = 24

∴ The maximum height is 24 feet

The arrow will go 24 feet to reach its maximum height and it takes 2 seconds to reach this height