a child flies a kite at a height of 70 ft, the wind carrying the kite horizontally away from the child at a…

Explanation:

Step1: Establish the relationship

Let the height of the kite be $y = 70$ ft (constant), the horizontal distance of the kite from the child be $x$, and the length of the string be $z$. By the Pythagorean - theorem, $x^{2}+y^{2}=z^{2}$. Since $y = 70$, we have $x^{2}+70^{2}=z^{2}$.

Step2: Differentiate with respect to time $t$

Differentiating both sides of the equation $x^{2}+4900 = z^{2}$ with respect to $t$ gives $2x\frac{dx}{dt}=2z\frac{dz}{dt}$, which simplifies to $x\frac{dx}{dt}=z\frac{dz}{dt}$.

Step3: Find the value of $x$ when $z = 250$

When $z = 250$, using $x^{2}+70^{2}=z^{2}$, we substitute $z = 250$ into it: $x^{2}+4900 = 250^{2}$, so $x^{2}=250^{2}-4900=62500 - 4900=57600$, and $x = 240$.

Step4: Solve for $\frac{dz}{dt}$

We know that $\frac{dx}{dt}=25$ ft/sec, $x = 240$, and $z = 250$. Substitute these values into $x\frac{dx}{dt}=z\frac{dz}{dt}$: $240\times25=250\times\frac{dz}{dt}$. Then $\frac{dz}{dt}=\frac{240\times25}{250}=24$ ft/sec.

Answer:

24