A certain clock has a minute hand that is 4 inches long and an hour hand is 3 inches long. How fast is the distance between the tips of the hands changing at 9:00?
let Ø be the angle between them
the angular velocity of the minute hand = 2π/60 rad/min = π/30 rad/min
the angular velicity of the hour hand = 2π/(12(60)) or π/360 rad/min
so dØ/dt = (π/30 - π.360) rad/min
= 11π/360 rad/min
let the distance between the tips be x inches
by Cosine Law
x^2 = 16 + 9 - 2(4)(3)cosØ
2x dx/dt = 24sinØ dØ/dt
when the time is 9:00, Ø = 90° and x = 5
10dx/dt = 24sin90° (11π/360)
x = 24(1)(11π/360)/10
= .2304 inches/min
this is wrong
This is right
this is wrong, right answer is 4.4pi in/h
To find the speed at which the distance between the tips of the hands is changing at 9:00, we need to consider the rates at which the minute and hour hands are moving.
At 9:00, the minute hand points at exactly 12, while the hour hand points at 9. This forms a right triangle between the center of the clock, the tip of the minute hand, and the tip of the hour hand.
Let's use the Pythagorean theorem to find the distance between the tips of the hands. The length of the side opposite the right angle is the difference between the lengths of the minute and hour hands, which is 4 - 3 = 1 inch. The length of the hypotenuse, which represents the distance between the tips of the hands, can be calculated as follows:
d = √(1^2 + (4 - 3)^2)
= √(1 + 1)
= √2
Therefore, at 9:00, the distance between the tips of the hands is √2 inches.
Now, in order to find how fast this distance is changing at 9:00, we need to consider the rates at which the minute and hour hands are moving. The minute hand moves 12 times faster than the hour hand because there are 12 segments on the clock face. Therefore, the rate at which the minute hand is moving is 12 times greater than the rate at which the hour hand is moving.
Since the length of the minute hand is 4 inches, and it takes 60 minutes to complete a full rotation, the rate at which the minute hand is moving can be calculated as:
(4 inches) / (60 minutes) = 2π/15 inches/minute
Similarly, since the length of the hour hand is 3 inches, and it takes 12 hours (720 minutes) to complete a full rotation, the rate at which the hour hand is moving can be calculated as:
(3 inches) / (720 minutes) = π/240 inches/minute
Now, using the chain rule of differentiation, we can find the rate at which the distance between the tips of the hands is changing at 9:00:
(d/dt)(d) = (d/dt)(√2) = (d/dt)(√(1 + 1)) = (1/2)(1 + 1)^(-1/2)(d/dt)(1 + 1)
(d/dt)(d) = (1/2)(d/dt)(2) = (1/2)(d/dt)(1) = 0
Therefore, at 9:00, the distance between the tips of the hands is not changing.