Question 177100
"write the height equation using this information"


Remember, the general height equation is {{{s=-16t^2+v[0]+s[0]}}} where "s" is the position of the object, "t" is the time, {{{v[0]}}} is the initial velocity, and {{{s[0]}}} is the initial position.



Since the "initial velocity of 32 feet per secound from the top of a 40 foot building", this means that {{{v[0]=32}}} and {{{s[0]=40}}}



So the equation is {{{s=-16t^2+32t+40}}} (after plugging in the initial velocity and position)



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b)


"how high is the rock after 5.0 seconds?"


Are you sure that the "5" isn't "0.5"? After 5 seconds, the rock wouldn't be in the air.




{{{s=-16t^2+32t+40}}} Start with the given equation.



{{{s=-16(0.5)^2+32(0.5)+40}}} Plug in {{{t=0.5}}}.



{{{s=-16(0.25)+32(0.5)+40}}} Square {{{0.5}}} to get {{{0.25}}}.



{{{s=-4+32(0.5)+40}}} Multiply {{{-16}}} and {{{0.25}}} to get {{{-4}}}.



{{{s=-4+16+40}}} Multiply {{{32}}} and {{{0.5}}} to get {{{16}}}.



{{{s=52}}} Combine like terms.



So after 0.5 seconds (half a second), the object is 52 ft in the air.



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c)



"after how many seconds will the rock reach maximum height"



To find the time where the object reaches the max height, we need to find the x-coordinate vertex of {{{s=-16t^2+32t+40}}}



{{{t=(-b)/(2a)}}} Start with the vertex formula.



From {{{s=-16t^2+32t+40}}}, we can see that {{{a=-16}}}, {{{b=32}}}, and {{{c=40}}}.



{{{t=(-(32))/(2(-16))}}} Plug in {{{a=-16}}} and {{{b=32}}}.



{{{t=(-32)/(-32)}}} Multiply 2 and {{{-16}}} to get {{{-32}}}.



{{{t=1}}} Divide.



So at one second, the object will reach the max height.



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d)


"what is the maximum height?"



{{{s=-16t^2+32t+40}}} Start with the given equation.



{{{s=-16(1)^2+32(1)+40}}} Plug in {{{t=1}}} (the time at which the object will reach the peak).



{{{s=-16(1)+32(1)+40}}} Square {{{1}}} to get {{{1}}}.



{{{s=-16+32(1)+40}}} Multiply {{{-16}}} and {{{1}}} to get {{{-16}}}.



{{{s=-16+32+40}}} Multiply {{{32}}} and {{{1}}} to get {{{32}}}.



{{{s=56}}} Combine like terms.



So the max height is 56 feet.