suppose a stone is thrown vertically upward from the edge of a cliff on a certain planet with an initial…

suppose a stone is thrown vertically upward from the edge of a cliff on a certain planet with an initial velocity of 50 ft/s from a height of 40 feet. the height s (in ft) of the stone abovethe ground t seconds after it is thrown is ( s = - 5 t ^ { 2 } + 50 t + 40 ). complete parts (a) through (f).\nthe height at the highest point is 165 feet.\n(type an exact answer, using radicals as needed.)\nd. when does the stone strike the ground?\nthe stone strikes the ground after ( 5 + sqrt { 33 } ) seconds.\n(type an exact answer, using radicals as needed.)\ne. with what velocity does the stone strike the ground?\nthe stone strikes the ground with a velocity of ( - 10 sqrt { 33 } ) ft/s.\n(type an exact answer using radicals as needed.)\nf. on what intervals is the speed increasing?\nthe speed is increasing on (square)\n(use interval notation simplify your answer. use integers or decimals for any numbers in the expression. round to two decimal places as needed.)

suppose a stone is thrown vertically upward from the edge of a cliff on a certain planet with an initial velocity of 50 ft/s from a height of 40 feet. the height s (in ft) of the stone abovethe ground t seconds after it is thrown is ( s = - 5 t ^ { 2 } + 50 t + 40 ). complete parts (a) through (f).\nthe height at the highest point is 165 feet.\n(type an exact answer, using radicals as needed.)\nd. when does the stone strike the ground?\nthe stone strikes the ground after ( 5 + sqrt { 33 } ) seconds.\n(type an exact answer, using radicals as needed.)\ne. with what velocity does the stone strike the ground?\nthe stone strikes the ground with a velocity of ( - 10 sqrt { 33 } ) ft/s.\n(type an exact answer using radicals as needed.)\nf. on what intervals is the speed increasing?\nthe speed is increasing on (square)\n(use interval notation simplify your answer. use integers or decimals for any numbers in the expression. round to two decimal places as needed.)

Answer

Explanation:

Step1: Find the derivative of the height function

The height function is (s(t)= - 5t^{2}+50t + 40). The velocity function (v(t)) is the derivative of (s(t)). Using the power rule ((x^{n})^\prime=nx^{n - 1}), we have (v(t)=s^\prime(t)=-10t + 50).

Step2: Analyze when the speed is increasing

Speed (|v(t)|). The acceleration (a(t)=v^\prime(t)=-10) (constant, since (v(t)=-10t + 50), and derivative of (-10t) is (-10) and derivative of (50) is (0)). The speed is increasing when the velocity and acceleration have the same sign. Since (a(t)=-10<0), we need to find when (v(t)<0). Set (v(t)=-10t + 50<0). Solve for (t): (-10t+50 < 0) (-10t<-50) (t > 5)

Answer:

((5,\infty))