The path of a cliff diver as he dives into a lake is given by the equation
y= -(x-10)^2+75, where y meters is the divers height above the water and x meters is the horizontal distance travelled by the diver. What is the maximum hight the diver is above water?
Is the answer to this question 75 meters? This is how I worked it out, please verify if correct:
parabola opens down
axis of symetry is 10 (h=10)
therefore y=-1(10-10)^2 + 75 = 75
Please is this right?
check with calculus
dy/dx = 2(x-10)
0 at max so x = 10
y = (10-10)^2 + 75
y = 75 sure enough
Many thanks
Yes, your calculation is correct. The maximum height the diver is above the water is 75 meters. The axis of symmetry of the parabolic equation is indeed at x = 10, and substituting this value into the equation gives y = -(10 - 10)^2 + 75 = 75.
Yes, you are correct! The maximum height the diver reaches above the water is indeed 75 meters.
To find the maximum height, you correctly identified that the parabola opens downward because of the negative coefficient of the squared term. The axis of symmetry, or the line that divides the parabola into two symmetric halves, is given by the value of x at the vertex of the parabola.
In this case, the equation of the parabola is y = -(x - 10)^2 + 75. From this equation, we can see that the vertex occurs when x - 10 equals zero, which means x = 10.
To find the maximum height, substitute the value of x = 10 into the equation: y = -(10 - 10)^2 + 75. Simplifying, we get y = -(0)^2 + 75, which is equal to 75.
So, the maximum height the diver reaches above the water is indeed 75 meters. Well done on working it out correctly!