Parabolas Application Q4

Question

A daring feat performed in Acapulco, Mexico is for a person to dive from a cliff into the ocean. Starting about 17 metres above the water, the height, h (in metres), of a diver t seconds after he jumps can be represented by the equation h=-4.9t²⁺1.5t+17.

 

a) Sketch a graph to represent the diver's height after jumping

b) How long does it take for the diver to reach the water ( to the nearest second)?

c) When does the diver reach his maximum height above the water. Give your answer correct to 2 decimal places.

d) What is the diver's greatest height above the water? Give your answer correct to the nearest cm.

 

I really need helps. Its parabola application and needs working out.

Answer 1

A quadratic function is in the form

 

y = ax² + bx + c

 

 

In this case, y=h    and   x=t.

 

 

a)

 

To sketch this parabola, we need to find 4 points: x-intercepts, y-intercept, and vertex.  To find the x-intercepts, we set the given function equal to zero.

 

0 = -4.9t² + 1.5t + 17

 

 

We can use the quadratic formula to solve for t. 

 

t = (-b ± √(b² - 4ac)) / 2a

 

where:
a= -4.9
b = 1.5
c = 17

 

Plug in these values into the formula.  You will have 2 values of t.   The coordinates of the x-intercepts will be  (t₁ , 0) and (t₂ , 0).

 

The y-intercept is the value of h when t=0.  The y-intercept is (0, 17).

 

 

The vertex has coordinates (j , k).  The vertex is the turning point of the parabola.  Use the formula

 

j = -b / (2a)  

 

to find the x coordinate of the vertex.  Then evaluate the given function at j to get the y coordinate of the vertex.

 

 

Once you have all the points, plot them on a coordinate system and connect them.  You should end up with a parabola the opens downward.

 

 

b)

 

Take the positive x-intercept that you calculated from part (a).

 

 

c and d)

 

Use  the vertex that you calculated from part (a).