Suppose that a particle has been shot into the air in such a way that the rate at which its height H (in feet per second) is changing is . If the particle is 3,100 feet high when seconds, write the formula for the height of the particle at any time t.

Question

Suppose that a particle has been shot into the air in such a way that the rate at which its height H (in feet per second) is changing is  . If the particle is 3,100 feet high when  seconds, write the formula for the height of the particle at any time t.

Quizplus · Accepted Answer

The given information is the rate of change of height, which is the derivative of the height function. Therefore, we need to find the antiderivative (or integral) of the given function to obtain the height function. Integrating 11ea984b_b163_5f08_9029_d779edb0c9fe_TB4005_11 with respect to t gives 11ea984b_b163_ad2b_9029_a9e1a827d599_TB4005_11 + C, where C is a constant of integration. We can find the value of C using the fact that the particle is 3,100 feet high when t=11ea984b_b163_8619_9029_95da000015c7_TB4005_11, so we plug in t=11ea984b_b163_8619_9029_95da000015c7_TB4005_11 and H=3,100 into the height function: 3,100 = 11ea984b_b163_ad2b_9029_a9e1a827d599_TB4005_11(11ea984b_b163_8619_9029_95da000015c7_TB4005_11) + C. Solving for C, we get C=3,100 - 11ea984b_b163_ad2b_9029_a9e1a827d599_TB4005_11(11ea984b_b163_8619_9029_95da000015c7_TB4005_11) = 11ea984b_b163_d43c_9029_9bd1eeb26232_TB4005_11. Therefore, the height function is H(t) = 11ea984b_b163_ad2b_9029_a9e1a827d599_TB4005_11t + 11ea984b_b163_d43c_9029_9bd1eeb26232_TB4005_11, which matches choice B.

Suppose That a Particle Has Been Shot into the Air