91Ó°ÊÓ

The Pythagorean theorem is a fundamental principle in geometry. It applies to right-angled triangles, stating that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In our ladder problem, the ladder acts as the hypotenuse of a right-angled triangle. The height of the ladder against the house forms one leg, and the distance between the base of the ladder and the wall forms the other leg. Using the equation: o[ x^2 + y^2 = z^2 The length of the ladder (hypotenuse) z is given as 17 ft. By plugging in x = 8 ft and solving for y, we can determine the height y of the ladder on the wall. Simply substituting the values, we get: x^2 + y^2 = 17^2 8^2 + y^2 = 17^264 + y^2 = 289 y^2 = 225 y = 15. This gives us the height of 15 ft. The Pythagorean theorem helps us understand how different sides of right-angled triangles relate, which is crucial in solving related rates problems.'

Differentiation with respect to time is a technique in calculus used to determine how a certain quantity changes over time. In our ladder problem, we have the relationship between the height y of the ladder on the wall and the distance x from the base of the ladder to the wall. As time t changes, both x and y change as well. oTo find how the height y changes as x changes over time, we differentiate both sides of the Pythagorean theorem with respect to t. The Pythagorean equation is y^2 + x^2 = 17^2. By differentiating each term with respect to time t, we get: 2y(dy/dt) + 2x(dx/dt) = 0. Knowing the rate at which x changes (dx/dt), we can solve this equation for dy/dt, the rate at which y changes. Plugging in the known values: 2(15)(dy/dt) + 2(8)(2) = 030(dy/dt) + 32 = 030(dy/dt) = -32dy/dt = -1.067 ft/sec. This shows the height of the ladder decreases by approximately 1.067 feet per second as the base slides away.'

Angle rate change refers to the rate at which an angle changes over time. In the ladder problem, we are interested in the angle θ between the ladder and the ground. To find the rate of change dθ/dt of this angle, we first express the angle in terms of the triangle's sides. We use the tangent function: tan(θ) = y/x. Differentiating both sides with respect to time t, we get: sec^2(θ) (dθ/dt) = (x(dy/dt) - y(dx/dt)) / x^2. Plugging in the known values x = 8, y = 15, dy/dt = -1.067, and dx/dt = 2, we proceed step by step to find dθ/dt. First, calculate sec(θ): sec(θ) = hypotenuse/adjacent = 17/8 sec^2(θ) = (17/8)^2 = 289/64. Substitute back into the differentiated equation: sec^2(θ) (dθ/dt) = (8(-1.067) - 15(2)) / 8^2 = ( -8.536 - 30 ) / 64 = -38.536 / 64. We have sec^2(θ) (dθ/dt) = -0.602. Solving for dθ/dt: dθ/dt = -0.602 / (289/64) = -0.602 * (64/289) = -0.133 rad/sec. This result shows the angle θ between the ladder and ground is decreasing at a rate of 0.133 radians per second.'

Related rates problems involve finding the rates at which related quantities change. In our ladder problem, as the base of the ladder slides away from the wall, different quantities like the height of the ladder along the wall and the angle between the ladder and the ground change over time. Here's how we can solve such problems: Identify the related quantities and their relationships. Specifically, use geometric or trigonometric relationships like the Pythagorean theorem or trigonometric functions. Differentiate these relationships with respect to time t. Substitute the given values for the quantities and their rates of change. Solve for the unknown rate. oIn our ladder problem: The Pythagorean theorem relates the height y and base x of the ladder to its constant length. Differentiate to find dy/dt given dx/dt. Use trigonometric functions to express the change in angle θ, differentiating to find dθ/dt. By following these steps, we systematically solve for how various related quantities change over time. This method is powerful for understanding dynamic systems in geometry and real-life applications.'