A platform diver follows a path determined by the equation h = −0.5d^2 + 2d + 6, where h represents
the height of the diver above the water and d represents the distance from the diving board. Both pronumerals are measured in metres.
Use the graph to determine:
a. how far the diver landed from the edge of the diving board
b. how high the diving board is above the water.
so, did you make the graph?
(a) where does h=0? The value of d there is how far she went
(b) find the vertex of the parabola.
To determine how far the diver landed from the edge of the diving board, we need to find the x-coordinate at which the height reaches zero. This represents the distance from the diving board at which the diver lands.
To solve for this, we need to set the equation equal to zero and solve for d:
0 = -0.5d^2 + 2d + 6
We can rearrange the equation to quadratic form:
0.5d^2 - 2d - 6 = 0
Next, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula:
d = (-b ± √(b^2 - 4ac))/(2a)
In our equation, a = 0.5, b = -2, and c = -6, so substituting the values:
d = (-(-2) ± √((-2)^2 - 4(0.5)(-6)))/(2(0.5))
Simplifying further:
d = (2 ± √(4 + 12))/(1)
d = (2 ± √(16))/(1)
d = (2 ± 4)/(1)
Now, we have two possible answers for d:
d₁ = (2 + 4)/(1) = 6
d₂ = (2 - 4)/(1) = -2
Since distance cannot be negative, we discard d₂ as an extraneous solution. Therefore, the diver landed 6 meters from the edge of the diving board.
To determine the height of the diving board above the water, we need to find the y-coordinate at which the graph intersects the y-axis. This represents the height of the diving board.
From the given equation, we can see that when d = 0, h = 6.
Therefore, the diving board is 6 meters above the water.