Exercise 30 Page 557

In this exercise, we are assigned three tasks given some initial information about geysers. The most relevant mathematical information is the initial upward velocity of 69 ft/sec. Let's consider the questions.

• What is the maximum height?
• What is the initial height?
• What is the function?

For convenience, we will find the initial height first, use that for our function, and, then, use that function to find the maximum height.

Initial Height

Knowing what a geyser is, helps us know the initial height. Geysers are essentially water fountains that spurt water from the ground naturally. This means the initial height of a geyser is the ground level of 0.

Function

The general formula for vertical motion is h=-16t^2+ vt+c, where v represents upward velocity and c represents the initial height of the object. In our case, the upward velocity is 69 ft/s and the initial height we just determined is . cc h&=&-16t^2&+& 69t&+& [0.8em] &&or [0.8em] h&=&-16t^2&+& 69t

Maximum Height

Now we can use the function to find the maximum. We will use the axis of symmetry to find the vertex. The h-value of the vertex is the maximum height of the function. We have a formula for the axis of symmetry of a quadratic. x=- b/2 a In our case, we can substitute a= -16 and b= 69 into the formula for the axis of symmetry. Instead of x, we use t.

t=- b/2 a

SubstituteII

a= -16, b= 69

t=-( 69)/2( -16)

▼

Calculate t

Multiply

Multiply

t=-69/-32

CalcQuot

Calculate quotient

t=2.15625

The axis of symmetry gives us the t-value of the vertex, t=2.15625. We can substitute that into our function to find the h-value.

h=-16t^2+69t

Substitute

x= 2.15625

h=( 2.15625)^2+69( 2.15625)

▼

Calculate h

CalcPow

Calculate power

h≈ 4.4694+69(2.15625)

Multiply

Multiply

h≈ 4.46941+148.78125

AddSubTerms

Add and subtract terms

h≈ 153.2566

The exercise suggests we round our answer to the nearest foot. We can now say that the maximum height of the geyser is 153ft.