This is a question regarding quadratic equations. I think I have to use the quadratic formula but I'm not sure and because the question provides the base I think I'm wrong. Here is the question|:
During a hurricane, parabolic waves on the ocean can exceed a height of 30 m above sea level. A ship hits a wave that rises 30 m above sea level and is 40 m wide at its base. If the wave hits the ship at 10 m/s, for how long will the ship be greater than 10 m above sea level?
if we set the wave with its vertex at (0,30) and it has height 0 at x=-20 and 20.
y = 30-ax^2
y(20) = 0, so 30/400 = a = 0.075
y = 30 - .075x^2
Now we want to know where y > 10:
30 - .075x^2 > 10
20 > .075x^2
-16.32 < x < 16.32
So, to travel a distance of 32.64m at 10 m/s requires 3.264 seconds.
See
http://www.wolframalpha.com/input/?i=plot+y%3D30+-+.075x%5E2%2C+y%3D10
To solve this problem, we can use the concept of projectile motion and quadratic equations. However, the quadratic formula is not required in this case.
We can start by analyzing the wave's shape and understanding its height as a function of time. Since the wave is described as parabolic, we can assume that its height follows a quadratic equation.
Let's consider a coordinate system where the wave's base is at the x-axis, and the y-axis represents the height above sea level. The peak height of the wave is 30 meters, and its base is 40 meters wide.
We can define the equation of the wave as h(x) = ax^2 + bx + c, where h(x) represents the height at a given distance x from the base.
Since the wave is symmetric, we know that at the base of the wave (x=0) and at the wave's widest point (x=20), the height is 0 meters. Therefore, we have two known points: (0,0) and (20,0).
Using these points, we can determine the equation of the wave. Substitute the values into the equation h(x) = ax^2 + bx + c to get two equations:
0 = a(0)^2 + b(0) + c --> c = 0
0 = a(20)^2 + b(20) + 0 --> 20a + b = 0 --> b = -20a
Now we have the equation h(x) = ax^2 + (-20a)x.
To find the value of 'a', we can use the given information that the peak height of the wave is 30 meters. Substituting the x-value of the peak into the equation, we have:
30 = a(10)^2 + (-20a)(10)
30 = 100a - 200a
-170a = 30
a = -30/170 = -3/17
Therefore, the equation of the wave is h(x) = (-3/17)x^2 + (60/17)x.
Now that we have the equation of the wave, we can determine for how long the ship will be greater than 10 meters above sea level. Since the height of the ship is greater than 10 meters above sea level, we need to solve the equation:
(-3/17)x^2 + (60/17)x > 10
To solve this quadratic inequality, we need to find its solutions. However, since the equation is not in standard form (with the inequality facing the right direction), we can rearrange it:
(-3/17)x^2 + (60/17)x - 10 > 0
Now we have a quadratic equation in standard form. To solve it, we can use methods like factoring, completing the square, or the quadratic formula.
Once we find the solutions of the quadratic equation, we can determine the time duration for which the ship will be greater than 10 meters above sea level by considering the positive region of the graph that lies between these solutions.
Note that due to the complexity of this quadratic inequality, the calculations may involve decimals or fractions.