Chapter 9.1, Problem 58PPS

a.

To determine

To graph the given equation.

Explanation of Solution

Given information :

Given function is $y=-16t^2+66t$ .

Graph :

Interpretation :

The graph of a quadratic equation ( $y=a{x}^2+bx+c$ ) is a parabola. The parabola is symmetric in nature and the point of symmetry is known as the vertex. This vertex lies on the line of symmetry and is symmetric about this line.

This vertex point shall be:

Highest point (if $a<0$ ) and be called the maximum. Here, the graph opens downwards.

Or, lowest point (if $a>0$ ) and can be called the minimum. Here the graph opens upwards.

In this case, ‘a’ is lesser than 0 hence the graph will have a maximum and will open downwards.

A parabola always points to infinity, either negative or positive.

To graph a quadratic function, compute the axis of symmetry, vertex and y-intercept, post which, plot the same on a graph.

The axis of symmetry bisects the parabola into two equal parts. Hence each point on the parabola would have an equal point on the other side of the axis of symmetry. Plot these points on the graph with a smooth curve.

Formula to compute equation of the axis of symmetry $x=-\frac{b}{2\times a}$ . Here, $a=-16$ and $b=66$ . ‘a’ and ‘b’ are coefficients $x^2$ and $x$ respectively.

Axis of symmetry for the given function is

  $x=-\frac{b}{2\times a}$

Formula for axis of symmetry.

  $x=-\frac{66}{2\times \left(-16\right)}$

Putting the values of ‘a’ and ‘b’ .

  $=\frac{66}{32}$

Simplifying this

  $x=\frac{33}{16}or,2.0625$ is the equation of axis of symmetry.

Vertex can be found out by putting the value of x computed in the axis of symmetry in the original function. This will give a value of y . These two coordinates of x and y would be the point where the vertex is.

  $=-16{\left(2.0625\right)}^2+66\left(2.0625\right)$

Putting the value of $x=2.0625$ in the original function.

  $=-68.0625+136.125$

Simplifying the expression.

  $=68.0625$

Thus the vertex is $\left(2.0625,68.0625\right)$ .

y-intercept is computed by substituting the value of x in the equation by 0.

  $=-16{\left(0\right)}^2+66\left(0\right)$

Simplifying

  $=0$

Hence the point of y-intercept is $\left(0,0\right)$ .

Now, plot these points along with their reflecting symmetric points, starting from the vertex.

b.

To determine

To calculate the maximum distance that M travelled north.

Answer to Problem 58PPS

The maximum distance that M travelled north is $68.0625$ feet .

Explanation of Solution

Given information :

The function provided is $y=-16t^2+66t$ .

Formula used :

The maximum distance travelled north can be calculated using the vertex formula. Here, find the y-coordinate of the point of the vertex. This value will be the maximum height attained by the M.

Formula to compute equation of the axis of symmetry $x=-\frac{b}{2\times a}$ . Here, $a=-16$ and $b=66$ . ‘a’ and ‘b’ are coefficients $x^2$ and $x$ respectively. Use this value in the original function to derive the maximum distance travelled.

Calculation :

  $x=-\frac{b}{2\times a}$

Formula for axis of symmetry.

  $x=-\frac{66}{2\times \left(-16\right)}$

Putting the values of ‘a’ and ‘b’ .

  $=\frac{66}{32}$

Simplifying this

  $x=\frac{33}{16}or,2.0625$ is the equation of axis of symmetry.

Vertex can be found out by putting the value of x computed in the axis of symmetry in the original function. This will give a value of y . These two coordinates of x and y would be the point where the vertex is.

  $=-16{\left(2.0625\right)}^2+66\left(2.0625\right)$

Putting the value of $x=2.0625$ in the original function.

  $=-68.0625+136.125$

Simplifying the expression.

  $=68.0625$

Thus the maximum distance travelled north is 68.0625 feet .

c.

To determine

To calculate the time taken to reach Casper Marina.

Answer to Problem 58PPS

Time taken to reach Casper Marina is $4.125$ minutes.

Explanation of Solution

Given information :

The function provided is $y=-16t^2+66t$ .

Formula used :

To calculate the total time taken, use the formula to find roots of a quadratic equation. This can be found by using $r=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ . Here ‘a’, ‘b’ and ‘c’ are the co-efficient of the given equation.

Here, $a=-16$ , $b=66$ and $c=0$ .

Calculation :

  $r=\frac{-66\pm \sqrt{{66}^2-4\times -16\times 0}}{2\times \left(-16\right)}$

Putting the values in the formula.

  $=\frac{-66\pm \sqrt{{66}^2}}{-32}$

Simplifying

  $=\frac{-66\pm 66}{-32}$

Removing square root.

  $=\frac{-66+66}{-32}$

One of the roots simplifications.

  $=0$

This is the first root.

  $=\frac{-66-66}{-32}$

Second root’s simplification.

  $=\frac{132}{32}$

Dividing the fraction.

  $=4.125$

Thus, M reached Casper Marina in 4.125 minutes .