A projectile is launched from a platform. Its height h(t), in meters, t seconds after laun… — SAT

A projectile is launched from a platform. Its height $h(t)$ , in meters, $t$ seconds after launch is modeled by the function $h(t) = -5t^2 + bt + 12$ , where $b$ is a positive constant. If the projectile reaches a maximum height of $32$ meters, what is the value of $b$ ?

Cevap: 20

Cevap

The correct answer is 20.

To find the maximum height of the projectile, we locate the vertex of the quadratic function

h(t) = -5t^2 + bt + 12

. The

t

-coordinate of the vertex is given by

t = -\frac{B}{2A} = -\frac{b}{2(-5)} = \frac{b}{10}

. Substituting

t = \frac{b}{10}

into

h(t)

and setting the height to

32

meters gives

-5\left(\frac{b}{10}\right)^2 + b\left(\frac{b}{10}\right) + 12 = 32

. Simplifying this equation results in

-\frac{b^2}{20} + \frac{b^2}{10} = 20

, which reduces to

\frac{b^2}{20} = 20

. Solving for

b

gives

b^2 = 400

. Because

b

must be a positive constant,

b = 20

Adım Adım Çözüm

Identify the time

t

at which the maximum height occurs using the vertex formula

t = -\frac{B}{2A}

for the quadratic function

h(t) = At^2 + Bt + C

t = \frac{b}{10}

The vertex of a downward-opening parabola represents its maximum value. For

h(t) = -5t^2 + bt + 12

, the coefficients are

A = -5

and

B = b

Substitute the time

t = \frac{b}{10}

back into the height function

h(t)

and set the expression equal to the maximum height of

32

meters.

-5\left(\frac{b}{10}\right)^2 + b\left(\frac{b}{10}\right) + 12 = 32

At the maximum height, the height of the projectile is

32

meters, which corresponds to the value of the function at the vertex time.

Simplify the equation to solve for

b^2

\frac{b^2}{20} = 20

Squaring the fraction yields

-5\left(\frac{b^2}{100}\right) + \frac{b^2}{10} + 12 = 32

. Simplifying the coefficients leads to

-\frac{b^2}{20} + \frac{b^2}{10} = 20

, which simplifies to

\frac{b^2}{20} = 20

Solve the equation for

b

b = 20

Multiplying both sides by

20

gives

b^2 = 400

. Taking the square root of both sides gives

b = \pm 20

. Since

b

is specified to be a positive constant,

b = 20

Anahtar Kavram

Finding the vertex of a quadratic function to determine maximum values in a real-world context.

Tahmini Süre:2m 0s

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