1) To determine the height of the wall, Lara needs to find the maximum value of the function h(t) = -0.81t^2 + 5t. The maximum value occurs at the vertex of the parabolic function, which can be found using the formula t = -b/2a.
In this case, a = -0.81 and b = 5. Plugging these values into the formula, we have:
t = -5 / (2 * -0.81)
t = 3.0864
Now, we can find the maximum height by substituting this value of t back into the function:
h(t) = -0.81(3.0864)^2 + 5(3.0864)
h(t) = 4.938
Therefore, the height of the wall is approximately 4.938 units.
2) To determine the distance between Harrison's cell and the town wall, Lara needs to find the horizontal distance traveled by the stone before it cleared the wall. This can be calculated by finding the total time it took for the stone to reach its maximum height and fall back down to the ground.
The stone reaches its maximum height at t = 3.0864, and it will reach the ground again when h(t) = 0. Solving for t in the equation -0.81t^2 + 5t = 0, we get:
t(-0.81t + 5) = 0
t = 0 or t = 5/0.81
t = 0 or t ≈ 6.1728
Since the time cannot be negative, we take t = 6.1728 as the time it takes for the stone to fall back to the ground.
Now we can find the horizontal distance traveled using the formula d = v * t, where v is the initial horizontal velocity of the stone.
The initial horizontal velocity can be found by considering the horizontal component of the initial velocity when Harrison launched the stone. Assuming the stone was launched horizontally, the initial velocity in the x-direction is 0. Therefore, the horizontal distance is given by:
d = 0 * 6.1728
d = 0
This means that Harrison's cell is right next to the town wall, with no horizontal distance between them.
Harrison and Lara—two tireless adventurers who face mathematical roadblocks and puzzles wherever they go—have a new challenge: 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.
1) 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.
2) How far is Harrison's cell from the town wall?
1 answer
1 answer