The arch that supports a highway bridge that passes over a street forms a parabola whose height above the street in feet can be determined using the function h(x)= -2x2+12.
a) How wide is the street?
b) Could a truck that is 10 feet tall be able to pass under the bridge?
I am abosolutely stumped on this question. Would anyone be willing to guide me?
(a) the width of the street is the distance between the roots (where h=0)
(b) assuming a zero-width truck, check the vertex of the parabola, which is, of course, midway between the roots, at x=0
Did you at least draw the graph?
Oh! Yes, I did. I just couldn't wrap my head around the question, I guess.
How would I find the distance between the roots? Would I need to find then first?
Of course! I'll be happy to guide you through this question.
To find the width of the street, we need to determine the value of x where the height of the bridge is equal to zero. In other words, we need to solve the equation h(x) = 0.
Given that the height of the bridge is given by the function h(x) = -2x^2 + 12, we set h(x) equal to zero and solve for x:
0 = -2x^2 + 12
To solve this quadratic equation, we can rearrange it to the standard form:
2x^2 = 12
Dividing both sides of the equation by 2 gives:
x^2 = 6
Taking the square root of both sides gives:
x = ±√6
Since we are dealing with a real-world situation, we can discard the negative solution. So the width of the street is approximately equal to √6.
Now, to determine if a truck that is 10 feet tall can pass under the bridge, we need to compare the height of the truck to the height of the bridge at its narrowest point.
The narrowest point of the bridge is located at the maximum of the parabolic function given by h(x). To find this maximum, we can use the vertex formula for quadratic functions:
x = -b / (2a)
In our case, a = -2 and b = 0, so the vertex can be found by:
x = -0 / (2 * -2)
x = 0
Substituting x = 0 into the equation h(x) = -2x^2 + 12 gives us the maximum height of the bridge:
h(0) = -2(0)^2 + 12
h(0) = 12
The maximum height of the bridge is 12 feet. Since the truck is 10 feet tall, it can pass under the bridge without any problems.
Therefore:
a) The width of the street is approximately equal to √6.
b) Yes, a truck that is 10 feet tall can pass under the bridge.