Question
A grasshopper leaps into the air from the edge of a cliff at a 50 degree angle. He reaches a maximum height 6.74 cm above the top of the cliff and travels a total horizontal distance of 1.06 m.
1) How tall is the cliff?
2) What is the initial speed of the grasshopper?
1) How tall is the cliff?
2) What is the initial speed of the grasshopper?
Answers
y = a(x-h)^2 + k
y'(0) = tan 50° = 1.1917
k = 0.0674
a(1.06-h)^2 + k = 0
2a(-h) = 1.1917
y = -0.7782(x-0.7657)^2+0.0674
y = -0.7782x^2 + 1.1917x - 0.3887
Let the height y be given by
y = ax^2+bx+c
y(-b/2a) = 0.0674
y(1.06) = 0
y'(0) = 2ax+b = tan 50° = 1.1917
so, we have
b = 1.1917
a(-1.1917/(2a))^2 + 1.1917(-1.1917/(2a))+c = 0.0674
1.1236a+1.1917*1.06 + c = 0
-0.3550/a + c = 0.0674
1.1236a + c = -1.2632
a = -0.7783
c = -0.3887
y = -0.7783x^2 + 1.1917x - 0.3887
DANG IT!
I get the same solution, and while the vertex and x-intercept are correct, I have y<0 at x=0, which is not right.
Maybe you can find my error.
y'(0) = tan 50° = 1.1917
k = 0.0674
a(1.06-h)^2 + k = 0
2a(-h) = 1.1917
y = -0.7782(x-0.7657)^2+0.0674
y = -0.7782x^2 + 1.1917x - 0.3887
Let the height y be given by
y = ax^2+bx+c
y(-b/2a) = 0.0674
y(1.06) = 0
y'(0) = 2ax+b = tan 50° = 1.1917
so, we have
b = 1.1917
a(-1.1917/(2a))^2 + 1.1917(-1.1917/(2a))+c = 0.0674
1.1236a+1.1917*1.06 + c = 0
-0.3550/a + c = 0.0674
1.1236a + c = -1.2632
a = -0.7783
c = -0.3887
y = -0.7783x^2 + 1.1917x - 0.3887
DANG IT!
I get the same solution, and while the vertex and x-intercept are correct, I have y<0 at x=0, which is not right.
Maybe you can find my error.