If the hands on a clocktower's clock are 2ft and 3ft long, how fast is the distance between the tips of the hands of the clock changing at 9:20?
I know that the derivative of the minute hand is pi/30 rad/min and derivative of the hour hand is pi/360 rad/min.
what is the equation i have to take the derivative of to find dtheta/dt?
starting at t=0 at noon, measured in minutes, the position (with (0,0) at the center of the clock) of the
hour hand is 2cos(2pi/720 t)
min. hand is 3cos(2pi/60 t)
so, the distance d between the hand tips at time t is
d^2 = 2^2 + 3^2 - 2*2*3 cos(pi*t/30 * 119/120)
2d dd/dt = 12(pi/30 * 119/120)sin(pi/30 * 119/120 t)
at 9:20, t=560, so
d = 3.663, so
dd/dt = 1/3.663 * 12(pi/30 * 119/120)sin(pi/30 * 119/120 560)
= 0.34 rad/min
Better check my math. The distance should be shrinking.
I'll try to revisit it later; maybe the above will give you a starting point, and you can fix my error.
I do see one error. The sign of the angle when figuring d:
pi/30 t(1/120 - 1) = -pi/30 t * 119/120, so that will change the sign of the dd/dt, so it is in fact shrinking.
in the first line of equation, where did "cos(pi*t/30 * 119/120)" come from?
the law of cosines uses the angle between the sides, which in our case is the angle between the hour hand and the minute hand.
since the equations are
hour hand is 2cos(2pi/720 t)
min. hand is 3cos(2pi/60 t)
the angle between the hands is
2pi/720 t - 2pi/60 t
= pi/360 t - pi/30 t
= pi/30 t * (1/120 - 1)
= pi/30 t * (-119/120)
oops, that would be pi/30 t (1/12 - 1) = pi/30 t * 11/12
that should change things a bit...
If you are 4\7mile from your house and you can walk 4 5\7 miles per hour how long will it take you to walk home?
I
To find the rate of change of the distance between the tips of the clock hands, you need to find the derivative of the equation that represents the distance between the tips of the clock hands with respect to time. Let's start by defining some variables:
- Let x represent the length of the shorter hand (2 ft).
- Let y represent the length of the longer hand (3 ft).
The distance between the tips of the clock hands can be calculated using the Law of Cosines, which states that:
c^2 = a^2 + b^2 - 2ab * cos(θ),
where a and b are the lengths of the two sides of a triangle, c is the length of the third side, and θ is the angle between the sides a and b.
In this case, a = x = 2 ft, b = y = 3 ft, and c represents the distance between the tips of the clock hands. The angle θ can be found as the difference between the positions of the two clock hands.
Now, let's express c in terms of x, y, and θ:
c^2 = x^2 + y^2 - 2xy * cos(θ).
To find the derivative of the distance c with respect to time t, we need to take the derivative of both sides of the equation:
d(c^2)/dt = d(x^2 + y^2 - 2xy * cos(θ))/dt.
Using the chain rule, we differentiate each term separately:
2c * dc/dt = 2x * dx/dt + 2y * dy/dt - 2xy * d(cos(θ))/dt.
Simplifying,
dc/dt = (x * dx/dt + y * dy/dt - xy * d(cos(θ))/dt) / c.
Now, substituting the given values for dx/dt and dy/dt, we have:
dc/dt = (x * (π/30) + y * (π/360) - xy * d(cos(θ))/dt) / c.
Now, we need to find the rate of change of cos(θ) with respect to time. Since θ represents the angle between the clock hands, and 9:20 can be visualized as an angle between the hour and minute hands, we can calculate θ by considering the current time on the clock.
In general, we can calculate θ as follows:
- Let h represent the number of hours and m represent the number of minutes.
- Each hour on the clock represents 30 degrees, and each minute represents 6 degrees.
- So, the current position of the hour hand can be given by θ_h = 30h + (m/2) degrees.
- The current position of the minute hand is given by θ_m = 6m degrees.
- Therefore, the angle between the two hands is θ = θ_m - θ_h.
By substituting the values for h=9 and m=20 into the above equation for θ, you can find its value, and then differentiate cos(θ) with respect to time to find d(cos(θ))/dt.
Finally, you can substitute all the given values into the equation for dc/dt to get the rate of change of the distance between the tips of the clock hands at 9:20.