Final answer:
The area of triangle ABC where BC=32 and tan B=3/2, and tan C=1/2 can be found by calculating the lengths of sides AB and AC using tangents, and then using the area formula, Area = 1/2 × base × height, where the base is 32 and the height is perpendicular to BC.
Step-by-step explanation:
To find the area of the triangle ABC with given values BC=32, tan B=3/2, and tan C=1/2, we must first find the lengths of the other sides using the tangent values and then apply the formula for the area of a triangle.
Since tan B = opposite/adjacent, we have AB/AC = 3/2. Similarly, for tan C = opposite/adjacent, we have AC/AB = 1/2. From these, AC = (2/3)AB and AB = 2(AC), allowing us to find the sides in terms of each other. With this relationship, we can use the Pythagorean theorem or the law of sines/cosines to find the actual lengths of AB and AC.
Next, we can use the area formula: Area = 1/2 × base × height. The base can be taken as side BC, but the height needs to be perpendicular to BC, which will be a segment we calculate after finding lengths AB and AC.
Upon finding the height, we multiply 1/2 by the base (32) by that height to get the area in square units.