A child is flying a kite. If the kite is 135 feet above the child's hand level and the wind is blowing it on a horizontal course at 7 feet per second, the child is paying out cord at ______ feet per second when 285 feet of cord are out. Assume that the cord remains straight from hand to kite.
I assume you have drawn a diagram of the situation. The length z of cord when the kite has flown to a horizontal distance x is
z^2 = x^2 + 135^2
so,
z dz/dt = x dx/dt
Find x when z=285, and you can find dz/dt.
To find the speed at which the child is paying out cord, we can use the concept of similar triangles.
Let's consider two triangles: Triangle ABC and Triangle ADE.
Triangle ABC represents the situation where the child is paying out the cord, with side AB representing the distance from the child's hand to the kite (135 feet), side BC representing the length of cord paid out (x feet), and angle BAC representing the angle of elevation of the kite.
Triangle ADE represents the situation after the cord has been paid out. Side AD represents the distance from the child's hand to the kite (135 + 285 = 420 feet), side DE represents the length of the cord paid out (285 feet), and angle DAE represents the angle of elevation of the kite.
Since the two triangles are similar, their corresponding sides are proportional:
AB / BC = AD / DE
Substituting the given values:
135 / x = 420 / 285
Cross-multiplying gives:
285 * 135 = 420 * x
38675 = 420 * x
x ≈ 92.27
Therefore, the child is paying out the cord at approximately 92.27 feet per second when 285 feet of the cord is out.
To find the rate at which the child is paying out cord, we can use the concept of similar triangles. Let's break down the problem into smaller parts.
First, we have two triangles: the triangle formed by the kite, the child's hand, and the string, and the larger triangle formed by the kite, the child's hand (which is at ground level), and the point directly below the kite.
Let's denote the height of the smaller triangle as "h" (135 feet), the length of the string as "x" (unknown), and the height of the larger triangle as "h + x."
Based on the problem statement, we can conclude that the two triangles are similar. This means that the ratio of their corresponding sides is equal.
Using this information, we can set up the following proportion:
h / x = (h + x) / 285
Now, let's solve for "x."
By cross-multiplying and simplifying the equation, we get:
h(285) = x(h + x)
135(285) = x(135 + x)
38675 = 135x + x^2
Rearranging the equation and simplifying, we get:
x^2 + 135x - 38675 = 0
At this point, we can use the quadratic formula to find the value of "x." After solving the quadratic equation, we find that "x" is approximately 165.62 feet.
Now that we know the length of the string, let's find the rate at which the child is paying out cord. We can use the concept of speed, which is equal to distance divided by time, to solve this.
The child is paying out cord at a rate of "y" feet per second, so we have the following equation:
y = x / t
We need to find the value of "t." The distance the kite travels horizontally is equal to the length of the string paid out, which is 285 feet. The kite moves at a constant speed of 7 feet per second, so we can set up the following equation:
285 = 7t
By solving for "t," we find that "t" is approximately 40.71 seconds.
Now, we can substitute the values of "x" (165.62 feet) and "t" (40.71 seconds) into the equation to find the rate at which the child is paying out cord:
y = 165.62 / 40.71
After performing the calculation, we find that the child is paying out cord at a rate of approximately 4.07 feet per second.