please help me solve this problem, thank you.
The ratio of the area of a triangle to the area of a rectangle is 3:5. the difference in their areas is 54 square inches. what is the height of the triangle if its base is 2 times its height?
5k - 3k = 54 ... k = 27
1/2 * 2h * h = 3 * 27
the height of the triangle is 9. Is that right
To solve this problem, we can follow these steps:
Step 1: Define the variables:
Let's define the variables:
- Let x be the height of the triangle.
- The base of the triangle would be 2x.
Step 2: Write the equation for the area of the triangle:
The area of a triangle can be calculated using the formula: 1/2 * base * height.
So, the area of the triangle in terms of x would be: 1/2 * (2x) * x = x^2.
Step 3: Write the equation for the area of the rectangle:
Let's denote the length and width of the rectangle as l and w, respectively.
The area of a rectangle is given by the formula: length * width.
So, the area of the rectangle in terms of l and w would be: l * w.
Step 4: Set up the given information as equations:
According to the problem statement, the ratio of the triangle's area to the rectangle's area is 3:5. Therefore, we have the equation:
x^2 / (lw) = 3/5.
Also, it is given that the difference in their areas is 54 square inches. Therefore, we have the equation:
x^2 - lw = 54.
Step 5: Solve the equations:
We now have a system of equations. We can substitute one equation into the other and solve for x.
From the first equation, we have:
lw = (5/3) * x^2.
Substituting this into the second equation, we have:
x^2 - (5/3) * x^2 = 54.
Simplifying this equation, we get:
(3/3 - 5/3) * x^2 = 54,
(-2/3) * x^2 = 54,
x^2 = -81.
However, since we are dealing with areas, x^2 cannot be negative. Therefore, there are no real solutions to this problem.
Thus, there is no valid answer for the height of the triangle that satisfies the given conditions.