Final answer:
The rate at which the area of the triangle changes cannot be determined without additional information, specifically the rate at which the base of the ladder is moving away from or towards the wall.
Step-by-step explanation:
To find the rate at which the area of a triangle formed by a ladder leaning against a wall is changing, we can use calculus. The area A of a right triangle with base b and height h can be given by A = (1/2)bh. In this case, the ladder serves as the hypotenuse (25 ft long), and we know one leg of the triangle (base b = 11 ft). First, we need to calculate the height (h) using the Pythagorean theorem:
c2 = a2 + b2
252 = h2 + 112
h2 = 252 - 112
h = √(252 - 112)
h = √(625 - 121)
h = √504
h = 22.47 feet (approximately)
Using the area formula A = (1/2)bh:
A = (1/2)(11 ft)(22.47 ft)
A = 123.59 square feet (approximately)
To find the rate of change of the area, we should differentiate the area with respect to time t (dA/dt). However, we're not given the rate at which the base is moving, so we cannot calculate the exact rate without it. Thus, with the information provided, we cannot determine the rate of change of area as a number.