Essay. Show all work. A gardener wants to create a triangular garden in the shape of a right triangle with the shortest side length x-3y ft. and the middle length side x+6y ft. What ia an algebraic expression for the area of the garden? be sure to multiply this out and express in simplest correct mathematical form, including units.
The "shortest length side" and the "middle length side" are the two perpendicular legs of a right triangle. For such a triangle, the area is 1/2 the product of the two shortest sides.
Now write that as an equation and do the rest.
You will be multiplying (x - 3y), (x + 6y), and a constant. The units will be square feet.
To find the area of a right triangle, we can use the formula A = (1/2) * base * height, where the base and height are the lengths of the two sides forming the right angle. In this case, the base will be the shorter side length x-3y, and the height will be the middle length side x+6y.
Therefore, the algebraic expression for the area of the garden is:
A = (1/2) * (x-3y) * (x+6y)
To simplify this expression, we need to distribute the terms. Multiplying (x-3y) by (x+6y) can be done using the distributive property, which states that a(b + c) = ab + ac.
So, applying the distributive property:
A = (1/2) * (x * x + x * 6y - 3y * x - 3y * 6y)
Next, we simplify:
A = (1/2) * (x^2 + 6xy - 3xy - 18y^2)
Combining the like terms, we get:
A = (1/2) * (x^2 + 3xy - 18y^2)
Finally, we can further simplify the expression by multiplying by (1/2):
A = 1/2 * x^2 + 1/2 * 3xy - 1/2 * 18y^2
Simplifying this further:
A = (1/2)x^2 + (3/2)xy - 9y^2
Therefore, the algebraic expression for the area of the triangular garden is (1/2)x^2 + (3/2)xy - 9y^2 square feet.