Question 456·Medium·Math

The mast and level ground meet at a right angle. From $A$ at the stake, the $33$-foot mast is opposite the angle. The $65$-foot wire is the hypotenuse, the side across from the right angle. The ground segment touches $A$ and is the adjacent leg; call its length $x$.

The Pythagorean theorem says the squares of the two legs add to the square of the hypotenuse. The ground and mast are the legs, so $x^2+33^2=65^2$. The wire belongs alone on the hypotenuse side, not with the legs.

Type $x_1^2+33^2\sim65^2\{x_1>0\}$. A regression uses $\sim$ to find a value that makes the sides equal; $x_1$ stands for the ground length $x$. The restriction keeps the length positive. Under PARAMETERS, Desmos shows $x_1=56$, so the stake is $56$ feet from the mast.

The tangent of an angle is its opposite leg divided by its adjacent leg. Here the mast is opposite $A$, and the ground length $x$ is adjacent, so $\tan A=\frac{33}{x}$. Tangent uses the two legs, not the wire.

Type $33/x_1$ below the regression. Desmos gives about $0.5892857$; tap the fraction button to see $\frac{33}{56}$. That's the tangent of the angle at the stake, not a length in feet. Choice A.