Find the number of seconds it will take to reach its maximum height. What is this maximum​ height?

If an object is projected upward from ground level with an initial velocity of 80 ft per​ sec, then its height in feet after t seconds is given by ​s(t)=−16t2+80t. Find the number of seconds it will take to reach its maximum height. What is this maximum​ height?

The object will take _____second(s) to reach its maximum height.

​(Simplify your​ answer.)

The maximum height reached by the object is _______ feet.

​(Simplify your​ answer.)

6 Answers

Relevance
  • 4 months ago
    Favorite Answer

    Method 1: By completing square

    s(t) = -16t² + 80t

    s(t) = -16[t² - 5t]

    s(t) = -16[t² - 5t + (2.5)²] + 16(2.5)²

    s(t) = -16[t - 2.5]² + 100

    For all real values of t, -16[t - 2.5]² ≤ 0

    Hence, s(t) = -16[t - 2.5]² + 100 ≤ 100

    Maximum s(t) = 100 at t = 2.5

    Answers:

    The object will take _2.5_ second(s) to reach its maximum height.

    The maximum height reached by the object is _100_ feet.

    ====

    Method 2: By differentiation

    s(t) = -16t² + 80t

    s'(t) = -32t + 80 = -32(t - 2.5)

    s"(t) = -32

    When t = 2.5:

    S(2.5) = -16(2.5)² + 80(2.5) = 100

    s'(2.5) = 0

    s"(2.5) = -32 < 0

    Hence, maximum s(t) = 100 at t = 2.5

    Answers:

    The object will take _2.5_ second(s) to reach its maximum height.

    The maximum height reached by the object is _100_ feet.

  • Jim
    Lv 7
    4 months ago

    Use -b/2a from Quadratic Equation (unless you know differentiation)

    t = -80/-32 s = 2.5 s will be max

    Then insert into original equation s(t)=−16(80/32)² +80(80/32)

    = 100 feet

    Attachment image
  • Vaman
    Lv 7
    4 months ago

    Velocity becomes at the highest point. ds/dt=-32 t+80. t= 80/32 s= 10/4=5/2 s. This is the time. The maximum height is h= 80*5/2-16*5/2*5/2= 200-4*25= 100 m.

  • ?
    Lv 7
    4 months ago

    The function

    .......... ​s(t) = −16t²+80t

    is a CONCAVE DOWN parabola. That means that its maximum height will occur at the parabola's vertex.

    The vertex occurs at t = -80/2(-16) = 2.5 seconds

    .................................s(t) = s(2.5) = 100 feet

    The object will take 2.5 second(s) to reach its maximum height......ANS

    The maximum height reached by the object is 100 feet...................ANS

    (​See graph below)

    Attachment image
  • How do you think about the answers? You can sign in to vote the answer.
  • 4 months ago

    There's a couple of ways to do this.  You could do the calculus method and find where the derivative is 0. But that;s not really necessary.  s(t) is a parabola, and the maximum height occurs at the vertex.

    The vertex occurs at -b/2a which is -80/(2(-16)) = 80/32 = 5/2 = 2.5 seconds.  Plug 2.5 into s(t) to get the height

    You could also have got the answer by finding the roots of the equation.  This one is easy because it factors

    into s(t) = t(-16t + 80).  This has roots when t = 0 and when -16t + 80 = 0, which happens when t = 5.

    The vertex is at the average of the roots.  so (0 + 5)/2 = 2.5

  • 4 months ago

    s(t) = -16t^2 + 80t

    The vertex of this parabola will provide information about maximum height.

    a = -16, b = 80, c =0 for the standard form of a quadratic equation.

    t = -b / 2a = -80 / 2(-16) = -80 / -32 = 2.5 seconds for max height.

    s(2.5) = -16(2.5^2) + 80(2.5) = 100 feet is max height attained.

Still have questions? Get your answers by asking now.