Harrison has been captured by a local warlord and is being held behind the very high walls of the warlord's walled town.
To scale the wall and rescue Harrison, Lara needs an ultra-light, carbon-fibre ladder. To minimize weight, it must be exactly the height of the walls and no higher.
Lara's only clue to the height of the wall is a secret message Harrison sent her, wrapped around a rock. Using a slingshot, he flung the rock over the wall from his cell window. Lara recognizes the message as a description of the arc that the stone made as it barely cleared the top of the wall. She can even tell how high the window was where Harrison held the slingshot (the Y in his sketch) to fire it.
Lara knows that Harrison would use t for time, and h(t) to represent the height of the stone at any particular moment in time. h(t) = −0.81t2 + 5t.
She can now reproduce Harrison's drawing to discover the height of the wall, as well as the distance between the wall and Harrison's cell window.
a.How tall a ladder does Lara need to reach the top of the wall? Don't forget that Harrison's equation describes the stone's flight from the cell window and barely over the wall.
b.How far is Harrison's cell from the town wall?
a. To find out the height of the wall, Lara needs to determine the maximum height reached by the stone. The equation h(t) = -0.81t^2 + 5t represents the height of the stone at any given time t.
Since the stone barely cleared the top of the wall, the maximum height it reached should be equal to the height of the wall. To find this maximum height, Lara can use the vertex formula for a parabola, which is given by t = -b / (2a).
In this case, a = -0.81 and b = 5. Substituting these values into the formula, we get:
t = -5 / (2*(-0.81))
t = -5 / (-1.62)
t ≈ 3.0864
So, the stone reaches its maximum height at approximately t = 3.0864 seconds.
Now, Lara can substitute this value back into the equation h(t) = -0.81t^2 + 5t to find the height of the wall:
h(3.0864) = -0.81(3.0864)^2 + 5(3.0864)
h ≈ 4.9997
Therefore, Lara needs an ultra-light, carbon-fibre ladder that is approximately 5 meters tall to reach the top of the wall.
b. To determine the distance between Harrison's cell window and the town wall, Lara needs to find the horizontal distance traveled by the stone before it reached the top of the wall.
Since the equation h(t) = -0.81t^2 + 5t represents the height of the stone at any given time t, Lara can find the time it took for the stone to reach the top of the wall by setting h(t) = 0 and solving for t:
-0.81t^2 + 5t = 0
t(-0.81t + 5) = 0
t = 0 (not relevant) or t = 6.17 (approx.)
Therefore, it took approximately 6.17 seconds for the stone to reach the top of the wall.
To find the horizontal distance traveled by the stone, Lara can use the equation d(t) = v0 * t, where v0 represents the initial horizontal velocity of the stone.
However, since the stone was flung over the wall, it had an initial horizontal velocity of zero. Therefore, the stone simply fell vertically and did not travel horizontally before reaching the top of the wall.
Hence, the distance between Harrison's cell window and the town wall is zero.
To find the height of the wall, we need to determine the maximum height reached by the stone, which corresponds to the height of the ladder needed.
a. To find the maximum height, we need to find the vertex of the quadratic equation h(t) = -0.81t^2 + 5t. The vertex of a quadratic equation in the form h(t) = at^2 + bt + c is given by the formula t = -b/(2a). Here, a = -0.81 and b = 5.
Let's calculate the time at which the stone reaches its maximum height:
t = -b/(2a) = -5/(2*(-0.81)) = 3.0864 seconds (rounded to four decimal places).
Next, substitute this value of t back into the equation h(t) to find the maximum height:
h(3.0864) = -0.81(3.0864)^2 + 5(3.0864) ≈ 7.8972 meters (rounded to four decimal places).
Therefore, Lara needs an approximately 7.8972-meter tall ladder to reach the top of the wall.
b. To find the distance between Harrison's cell and the town wall, we can calculate the horizontal distance traveled by the stone at the time it reaches the maximum height.
To find this distance, we need to use the equation d(t) = vt, where v is the initial velocity of the stone (which is not given but can be assumed as 0 since it was thrown straight up) and t is the time at which the stone reaches its maximum height (3.0864 seconds).
d(3.0864) = 0 * 3.0864 = 0 meters.
Based on the calculation, we can conclude that Harrison's cell is directly at the town wall, so there is no horizontal distance between them.