Chapter ISG, Problem 9.6.2P

a.

To determine

To find what is meaning of each term.

Answer to Problem 9.6.2P

a determines how wide or narrow the graphs are, and whether the graph turns upward or downward.

The linear-term coefficient b shifts the axis of symmetry away from the y-axis.

The constant term c affects the y-intercept. The greater the number, the higher the intercept point on the y -axis.

Explanation of Solution

Given information:

  $h_1=-16t^2+90t+120$

  $h_1=-16t^2+90t+120$

Here

  $a=-16$

  $b=90$

  $c=120$

Here, coefficient a is negative, therefore the endpoints of the parabola point downward.

The coefficient a , determines how wide or narrow the graphs are, and whether the graph turns upward or downward.

The linear-term coefficient b shifts the axis of symmetry away from the y-axis. The direction of shift depends on the sign of the quadratic coefficient and the sign of the linear coefficient.

Initial velocity is 90

The axis of symmetry shifts to the right because equation has negative a and positive b coefficients.

The constant term c affects the y-intercept. The greater the number, the higher the intercept point on the y -axis.

Initial height of the rocket is 120

b.

To determine

To find when rocket hit the rocket.

Answer to Problem 9.6.2P

  $t_1\approx 1.11$ seconds

  $t_2\approx 6.73$ seconds

Explanation of Solution

Given information:

  $h_1=-16t^2+90t+120$

Formula used:

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Calculation:

  $h_1=-16t^2+90t+120$

When rocket hit ground then $h_1=0$

  $0=-16t^2+90t+120$

On solving,

  $16t^2-90t-120=0$

  $a=16$

  $b=-90$

  $c=-120$

Using formula

  $t=\frac{-(-45)\pm \sqrt{{(-45)}^2-4(8)(-60)}}{2(8)}$

On solving,

  $t=\frac{45\pm \sqrt{2025+1920}}{16}$

On solving,

  $t=\frac{45\pm \sqrt{3945}}{16}$

On solving,

  $t_1=\frac{45+\sqrt{3945}}{16}$

  $t_2=\frac{45-\sqrt{3945}}{16}$

On solving,

  $t_1\approx 1.11$ seconds

  $t_2\approx 6.73$ seconds

c.

To determine

To find maximum height .

Answer to Problem 9.6.2P

  $h_1=240$ feet

  $t=2.81$ second attain maximum height

Explanation of Solution

Given information:

  $h_1=-16t^2+90t+120$

Calculation:

A parabola reaches its maximum value at its vertex, or turning point. Use the formula for the axis of symmetry to find the x -coordinate of the vertex.

  $t=\frac{-b}{2a}$

  $t=\frac{-90}{2\times -16}$

On solving,

  $t=2.81$ second attain maximum height

Maximum height

  $h_1=-16{(2.81)}^2+90(2.81)+120$

On solving,

  $h_1=-16\times 4.75+196.2+120$

On solving,

  $h_1=-76+196.2+120$

On solving,

  $h_1=240$ feet

d.

To determine

To explain shape of graph .

Answer to Problem 9.6.2P

Parabola

Explanation of Solution

Given information:

  $h_1=-16t^2+90t+120$

Notice that equation of height is a quadratic function, which means its graph will be a parabola.

e.

To determine

To find the value of coins .

Answer to Problem 9.6.2P

  $t_1\approx 1.106$

  $t_2\approx 4.51$

Explanation of Solution

Given information:

  $h_1=-16t^2+90t+120$

height is 200 feet

Calculation:

  $h_1=-16t^2+90t+120$

As $h_1=200$

  $200=-16t^2+90t+120$

On solving,

  $16t^2-90t+80=0$

On solving,

  $8t^2-45t+40=0$

On solving,

  $t=\frac{-(-45)\pm \sqrt{{(-45)}^2-4(8)(40)}}{2(8)}$

On solving,

  $t=\frac{45\pm \sqrt{2025-1280)}}{16}$

On solving,

  $t=\frac{45\pm \sqrt{745}}{16}$

On solving,

  $t=\frac{45+\sqrt{745}}{16}$

  $t=\frac{45-\sqrt{745}}{16}$

On solving,

  $t_1\approx 1.106$

  $t_2\approx 4.51$