Question

solve the problem. a kite is flying on 149 feet of string. how high is it above the ground if its height is 89 feet more than the horizontal distance from the person flying it? assume the string is being released at ground level. the kite is feet above the ground.

Explanation:

Step1: Apply Pythagorean theorem

Let the horizontal distance be $x$ and the height of the kite above the person be $x - 89$, and the length of the string be the hypotenuse $c = 149$. According to the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, so $(x - 89)^{2}+x^{2}=149^{2}$.

Step2: Expand the equation

Expand $(x - 89)^{2}$ using the formula $(a - b)^{2}=a^{2}-2ab + b^{2}$. We get $x^{2}-178x + 89^{2}+x^{2}=149^{2}$, which simplifies to $2x^{2}-178x+7921 = 22201$.

Step3: Rearrange to quadratic - form

Rearrange the equation to $2x^{2}-178x+7921 - 22201=0$, so $2x^{2}-178x - 14280 = 0$. Divide through by 2 to get $x^{2}-89x - 7140 = 0$.

Step4: Solve the quadratic equation

Use the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ for the quadratic equation $ax^{2}+bx + c = 0$. Here, $a = 1$, $b=-89$, and $c=-7140$. First, calculate the discriminant $\Delta=b^{2}-4ac=(-89)^{2}-4\times1\times(-7140)=7921 + 28560=36481$. Then $x=\frac{89\pm\sqrt{36481}}{2}=\frac{89\pm191}{2}$. We take the positive root $x=\frac{89 + 191}{2}=\frac{280}{2}=140$ (since distance can't be negative).

Step5: Find the height of the kite

The height of the kite above the ground is $x-89 + 89=x$. So the height of the kite above the ground is 140 feet.

Answer:

140