Final answer:
To find the dimensions of the rectangle of maximum area inscribed in a right triangle with legs of length 12 and 14, we can set up an equation for the area of the rectangle and solve for the dimensions. By taking the derivative of the area equation with respect to one of the dimensions, we can find the maximum area. Evaluating the area at the endpoints, we can determine the dimensions of the rectangle. Hence the correct answer is option D
Step-by-step explanation:
To find the dimensions of the rectangle of maximum area inscribed in a right triangle, we can start by drawing a diagram. Let's label the width of the rectangle as x and the height as y. Since the sides of the rectangle are parallel to the legs of the triangle, we can see that the width and height will be the legs a and b respectively. Therefore, the dimensions of the rectangle are a = 12 and b = 14.
Now, let's write the equation for the area of the rectangle in terms of x and y: A = xy. We also know that the sides of the rectangle are parallel to the legs of the triangle, so their product should be equal to the area of the triangle: xy = (1/2)ab = (1/2)(12)(14) = 84.
Next, let's solve this equation for y: y = 84/x. Now we can substitute this expression for y back into the equation for the area: A = x(84/x) = 84.
To find the maximum value of A, we can look for critical points by taking the derivative of A with respect to x and setting it equal to zero. The derivative of A with respect to x is A' = 84/x^2. Setting this equal to zero, we find that there are no critical points. Therefore, the maximum value of A occurs at the endpoints of the interval [0, 12]. Evaluating A at these endpoints, we can see that the maximum area is 84.
Since the dimensions of the rectangle are a = 12 and b = 14, the correct answer is option D) 84/5, 42/5.