A piece of elastic is attached to two nails on a flat board, with a button attached to the midpoint of the elastic. The ails are 5 cm apart. You stretch the elastic by pulling the button along the board i the direction that is perpendicular to the line between the nails.
A. Find the equation that relates the total length of the elastic x to the distance y that the button has moved.
B. You pull down the button at a constant 3cm/sec. Find the rate at which the length of the elastic is increasing when it is 12cm long.
each side of the elastic is the hypotenuse of a triangle with legs 5/2 and y, so
x = 2√(2.5^2 + y^2)
dx/dt = 2y/√(2.5^2+y^2) dy/dt
when x=12, y=5.45, so
dx/dt = 2(5.45)/6 (-3) = -1/5.45 = -0.18 cm/s
A.
x = 2√[ y^2 + (2.5)^2 ] cm (using Pythagorus theorem)
B.
x^2 = 4y^2 + 25
=> 2x dx/dt = 8y dy/dt
=> dx/dt = 4 (y/x) dy/dt
When x = 12 cm, y = (1/2)√[ 144 - 25 ] = 2.727 cm
=> dx/dt = 4 (2.727/12) x 3 cm/s
= 2.727 cm/s.
A. To find the equation that relates the total length of the elastic x to the distance y, we can use the Pythagorean theorem.
Let's label the distance from one nail to the button as a and the distance from the other nail to the button as b.
According to the Pythagorean theorem, a^2 + b^2 = c^2, where c is the hypotenuse (total length of the elastic).
Since the nails are 5 cm apart, a = b = 5/2 = 2.5 cm.
Now, let's express c in terms of y. As you pull the button along the board, the elastic will stretch, and the length of each side will increase. Therefore, a and b will both increase by the same amount, which we'll call dy.
So, a = 2.5 + dy and b = 2.5 + dy.
Plugging these values into the equation, we have:
(2.5 + dy)^2 + (2.5 + dy)^2 = c^2.
Simplifying, we get:
6.25 + 5dy + d^2y^2 + 6.25 + 5dy + d^2y^2 = c^2,
12.5 + 10dy + 2d^2y^2 = c^2.
Now, let's express c in terms of x. The total length of the elastic x is equal to twice the hypotenuse:
x = 2c,
c = x/2.
Plugging this back into our equation, we have:
12.5 + 10dy + 2d^2y^2 = (x/2)^2,
12.5 + 10dy + 2d^2y^2 = x^2/4.
Multiplying both sides by 4, we get:
50 + 40dy + 8d^2y^2 = x^2.
Therefore, the equation that relates the total length of the elastic x to the distance y that the button has moved is:
x^2 = 8d^2y^2 + 40dy + 50.
B. To find the rate at which the length of the elastic is increasing when it is 12 cm long, we need to find dx/dt when x = 12.
Differentiating both sides of the equation with respect to t, we get:
2x(dx/dt) = 16d^2y(dy/dt) + 40dy/dt.
Since we are given that dy/dt = 3 cm/sec, we can substitute this value into the equation.
Plugging in x = 12 and dy/dt = 3, we have:
2(12)(dx/dt) = 16d^2(0) + 40(3),
24(dx/dt) = 120,
dx/dt = 120/24,
dx/dt = 5 cm/sec.
Therefore, the rate at which the length of the elastic is increasing when it is 12 cm long is 5 cm/sec.
A. To find the equation that relates the total length of the elastic x to the distance y that the button has moved, we need to consider the shape of the elastic.
Since the nails are fixed points, we can consider the elastic as a line segment with length 2a, where a is the distance from each nail to the midpoint. According to the given information, the nails are 5 cm apart, so each nail is 2.5 cm away from the midpoint.
Let's assume that the midpoint is at the origin (0,0) on a coordinate plane. When the button is moved in the positive y-direction, it will increase the total length of the elastic. We can represent the position of the button as (0, y).
Since we are stretching the elastic, the length of the elastic in the positive y-direction will be a + y (distance from the origin to the button) and the length in the negative y-direction will be a - y (distance from the origin to the nail).
By utilizing the Pythagorean theorem, we can find the total length of the elastic:
(x/2)^2 = (a + y)^2 + (2.5)^2 (1)
(x/2)^2 = (a - y)^2 + (2.5)^2 (2)
Simplifying equation (1) and (2) will yield the equation that relates x and y.
B. To find the rate at which the length of the elastic is increasing when it is 12 cm long, we need to differentiate the equation from part A with respect to time and substitute the given values.
Let's assume x and y are functions of time t (x = x(t), y = y(t)).
Differentiating equation (1) and (2) with respect to t, we get:
[1/2 * (x/2)^2]' = [((a+y)^2 + (2.5)^2)]' (1)
[1/2 * (x/2)^2]' = [((a-y)^2 + (2.5)^2)]' (2)
Simplifying equation (1) and (2), we will obtain the derivatives:
x/2 * (dx/dt) = 2(a+y) * (dy/dt) (3)
x/2 * (dx/dt) = 2(a-y) * (-dy/dt) (4)
Given dx/dt (the rate at which the button is moving) as 3 cm/sec and x = 12 cm, and substituting the values into equation (3), we can find dy/dt (the rate at which the length of the elastic is increasing) when x = 12 cm.
Finally, substitute the values into equation (3) to solve for dy/dt, which will give us the rate at which the length of the elastic is increasing.