A right-triangle sail with an area but an unknown leg is a two-stage tangent question. First use the triangle’s area formula and its perpendicular legs to find the missing length; Desmos can solve that one-unknown equation. Then name opposite and adjacent from the marked angle before making the ratio. Reversing the legs gives the tangent of the other acute angle.
Hints
- Hint 1
The triangle area formula multiplies a base by a perpendicular height and halves the product. Since the mast meets the boom at a right angle, which leg’s length do you need to find?
- Hint 2
Once you write the area equation with the mast’s height as the one unknown, replace with in Desmos. That tells it to find the height that makes the given area work.
- Hint 3
At angle on the outer end of the boom, the mast is across from the angle and the boom touches it. Tangent divides the opposite leg by the adjacent leg. Which one belongs on top?
Step-by-step
Find the mast, then form the tangent ratio
Step 1Turn the sail’s area into an equation
Let be the mast’s height in feet. The mast meets the -foot boom at a right angle, so they are the perpendicular height and base in . Using the given area of square feet: . The slanted edge is not the height.
- Step 2
Find the mast’s length in Desmos
Type . The tells Desmos to find the unknown height that fits the equation. Under PARAMETERS, it shows , so the mast is feet tall.
- Step 3
Name the legs from angle x
The opposite leg sits across from an angle. From at the outer end of the boom, that’s the mast, feet long. The adjacent leg touches and isn’t the slanted side across from the right angle, so it’s the -foot boom.
- Step 4
Write the tangent ratio
Don’t put the boom on top: that reverses the ratio and answers for the other acute angle. For the sail’s angle, tangent is opposite over adjacent: . Choice B.