Question

An object is propelled from the ground with an initial upward velocity of 224 feet per second. Will the object reach a height of 784 feet? If it does, how long will it take the object to reach that height? Solve by factoring.

Short answer

Yes, the object does reach a height of 784 feet and it takes 7 seconds to reach that height.

Step-by-step solution

Formulate the equation

The equation is: \( 784 = 224t - 0.5*32t^2 \). This simplifies to \( 784 = 224t - 16t^2 \)

Set the equation to zero

We then rewrite this equation in the standard form for a quadratic equation (which is set to zero): \( 0 = 16t^2 - 224t + 784 \).

Factor the quadratic equation

Factoring the quadratic equation gives: \( 0 = 16(t - 7)(t - 7) \).

Solve for t

Setting each factor equal to zero gives the solutions: \( t = 7 \), \( t = 7 \). We have two equal solutions, so the object does reach the height of 784 feet and it takes 7 seconds to reach that height.