path of a frogs leap: a frog leaps from a box 3 feet high and lands 4 feet from the base of the stump. we can consider the initial position of the frog to be at (0,3) and its landing position to be at (4,0). it is determined that the height of the frog as a function of its horizontal distance x from the base of the box is given by: h(x)=-0.5x^2+1.25x+3 where x and h(x) are both in feet.
a)how high was the frog when its horizontal distances from the base of the box was 2 feet?
b)at what two horizontal distances from the base of the box was the frog 3.25 feet above the ground?
c)at what horizontal distance from the base of the box did the frog reach its highest point?
d) what was the maximum height reached by the frog?
h(x)=-0.5x^2+1.25x+3
h(2) = 3.5
3.25 = -0.5x^2+1.25x+3
-0.5x^2+1.25x-.25 = 0
x = .22 (going up) or 2.28 (coming down)
a parabola reaches its vertex when x = -b/2a = -1.25/-1 = 1.25
h(1.25) = 3.781
a) When the horizontal distance from the base of the box is 2 feet (x = 2), we can find the height of the frog by substituting x into the equation h(x) = -0.5x^2 + 1.25x + 3:
h(2) = -0.5(2)^2 + 1.25(2) + 3
= -0.5(4) + 2.5 + 3
= -2 + 2.5 + 3
= 3.5 feet
Therefore, the frog was 3.5 feet above the ground when its horizontal distance from the base of the box was 2 feet.
b) To find the horizontal distances from the base of the box when the frog was 3.25 feet above the ground (h(x) = 3.25), we need to solve the equation -0.5x^2 + 1.25x + 3 = 3.25:
-0.5x^2 + 1.25x + 3 - 3.25 = 0
-0.5x^2 + 1.25x - 0.25 = 0
Using the quadratic formula, we find the two horizontal distances:
x = (-1.25 ± √(1.25^2 - 4 * (-0.5) * (-0.25))) / (2 * (-0.5))
After simplification:
x = (-1.25 ± √(1.5625 - 0.5)) / (-1)
x = (-1.25 ± √(1.0625)) / (-1)
x = (-1.25 ± 1.031) / (-1)
This gives us two possible solutions:
x1 = (-1.25 + 1.031) / (-1) = -0.219 feet
x2 = (-1.25 - 1.031) / (-1) = 2.281 feet
Therefore, the frog was 3.25 feet above the ground at approximately -0.219 feet and 2.281 feet from the base of the box.
c) To find the horizontal distance from the base of the box when the frog reached its highest point, we can use the vertex formula:
x = -b / (2a)
From the equation h(x) = -0.5x^2 + 1.25x + 3, we can extract the values a = -0.5 and b = 1.25:
x = -(1.25) / (2 * (-0.5))
x = -1.25 / (-1)
x = 1.25 feet
Therefore, the frog reached its highest point at a horizontal distance of 1.25 feet from the base of the box.
d) To find the maximum height reached by the frog, we can substitute the x-coordinate of the highest point (x = 1.25) into the equation h(x) = -0.5x^2 + 1.25x + 3:
h(1.25) = -0.5(1.25)^2 + 1.25(1.25) + 3
= -0.5(1.5625) + 1.5625 + 3
= -0.78125 + 1.5625 + 3
= 3.78125 feet
Therefore, the maximum height reached by the frog is approximately 3.78125 feet.
a) To find the height of the frog when its horizontal distance from the base of the box is 2 feet, substitute x = 2 into the equation h(x).
h(x) = -0.5x^2 + 1.25x + 3
h(2) = -0.5(2)^2 + 1.25(2) + 3
h(2) = -0.5(4) + 2.5 + 3
h(2) = -2 + 2.5 + 3
h(2) = 3.5 feet
Therefore, when the frog is 2 feet from the base of the box, its height is 3.5 feet.
b) To find the horizontal distances from the base of the box when the frog is 3.25 feet above the ground, we need to solve the quadratic equation h(x) = 3.25.
-0.5x^2 + 1.25x + 3 = 3.25
Rearranging the equation:
-0.5x^2 + 1.25x - 0.25 = 0
Using the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
For this equation, a = -0.5, b = 1.25, and c = -0.25.
x = (-1.25 ± sqrt(1.25^2 - 4(-0.5)(-0.25))) / (2(-0.5))
x = (-1.25 ± sqrt(1.5625 - 0.5)) / (-1)
x = (-1.25 ± sqrt(1.0625)) / (-1)
x = (-1.25 ± 1.031) / (-1)
x1 = (-1.25 + 1.031) / (-1)
x1 = -0.219 / -1
x1 = 0.219 feet
x2 = (-1.25 - 1.031) / (-1)
x2 = -2.281 / -1
x2 = 2.281 feet
Therefore, the frog is 3.25 feet above the ground at the horizontal distances of 0.219 feet and 2.281 feet from the base of the box.
c) To find the horizontal distance from the base of the box where the frog reaches its highest point, we can find the vertex of the quadratic equation h(x) = -0.5x^2 + 1.25x + 3.
The x-coordinate of the vertex is given by:
x = -b / (2a)
For this equation, a = -0.5 and b = 1.25.
x = -1.25 / (2*(-0.5))
x = -1.25 / (-1)
x = 1.25 feet
Therefore, the frog reaches its highest point at a horizontal distance of 1.25 feet from the base of the box.
d) To find the maximum height reached by the frog, we can substitute the x-coordinate of the vertex into the equation h(x).
h(x) = -0.5x^2 + 1.25x + 3
h(1.25) = -0.5(1.25)^2 + 1.25(1.25) + 3
h(1.25) = -0.5(1.5625) + 1.5625 + 3
h(1.25) = -0.78125 + 1.5625 + 3
h(1.25) = 3.78125 feet
Therefore, the maximum height reached by the frog is 3.78125 feet.
To answer these questions, we need to substitute the given values of x into the function h(x) = -0.5x^2 + 1.25x + 3 and solve for the corresponding height values.
a) To find the height when the horizontal distance from the base of the box is 2 feet, substitute x = 2 into the equation:
h(2) = -0.5(2)^2 + 1.25(2) + 3
= -0.5(4) + 2.5 + 3
= -2 + 2.5 + 3
= 0.5 + 3
= 3.5
Therefore, when the frog is at a horizontal distance of 2 feet from the base of the box, it is at a height of 3.5 feet.
b) To find the horizontal distances when the frog is 3.25 feet above the ground, set h(x) = 3.25 and solve for x:
-0.5x^2 + 1.25x + 3 = 3.25
-0.5x^2 + 1.25x - 0.25 = 0
To solve this quadratic equation, you can either factor it or use the quadratic formula. Assuming we use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For this equation, a = -0.5, b = 1.25, and c = -0.25. Plugging these values into the formula:
x = (-1.25 ± √(1.25^2 - 4(-0.5)(-0.25))) / (2(-0.5))
Simplifying further, we get:
x = (-1.25 ± √(1.5625 - 0.5)) / (-1)
= (-1.25 ± √(1.0625)) / (-1)
= (-1.25 ± 1.03125) / (-1)
Now we can calculate the two possible values for x:
x1 = (-1.25 + 1.03125) / (-1)
= -0.21875 / (-1)
= 0.21875
x2 = (-1.25 - 1.03125) / (-1)
= -2.28125 / (-1)
= 2.28125
Therefore, the frog is 3.25 feet above the ground at the horizontal distances of approximately 0.21875 feet and 2.28125 feet from the base of the box.
c) To find the horizontal distance where the frog reaches its highest point, we need to find the vertex of the quadratic function. The x-coordinate of the vertex can be found using the formula:
x = -b / (2a)
For our equation, a = -0.5 and b = 1.25. Plugging in the values:
x = -1.25 / (2(-0.5))
= -1.25 / (-1)
= 1.25
Therefore, the frog reaches its highest point at a horizontal distance of 1.25 feet from the base of the box.
d) The maximum height reached by the frog can be found by substituting the x-coordinate of the vertex into the function h(x):
h(1.25) = -0.5(1.25)^2 + 1.25(1.25) + 3
= -0.5(1.5625) + 1.5625 + 3
= -0.78125 + 1.5625 + 3
= 0.78125 + 3
= 3.78125
Therefore, the maximum height reached by the frog is approximately 3.78125 feet.