Textbook Question
Diagonals If x, y, and z are lengths of the edges of a rectangular box, then the common length of the box’s diagonals is s = √(x² + y² + z²).
c. How are dx/dt, dy/dt, and dz/dt related if s is constant?
4. Applications of Derivatives
Related Rates
6:40 minutesProblem 3.8.23b
Textbook Question
A sliding ladder
A 13-ft ladder is leaning against a house when its base starts to slide away. By the time the base is 12 ft from the house, the base is moving at the rate of 5 ft/sec.
b. At what rate is the area of the triangle formed by the ladder, wall, and ground changing then?
Verified step by step guidance1
First, identify the triangle formed by the ladder, the wall, and the ground. The ladder acts as the hypotenuse, the distance from the wall to the base of the ladder is one leg, and the height from the ground to the top of the ladder is the other leg.
Use the Pythagorean theorem to relate the sides of the triangle: \( x^2 + y^2 = 13^2 \), where \( x \) is the distance from the wall to the base of the ladder, and \( y \) is the height of the ladder on the wall.
Differentiate the Pythagorean equation with respect to time \( t \) to find the relationship between the rates of change: \( 2x \frac{dx}{dt} + 2y \frac{dy}{dt} = 0 \). Given \( \frac{dx}{dt} = 5 \) ft/sec and \( x = 12 \) ft, solve for \( \frac{dy}{dt} \).
The area \( A \) of the triangle is given by \( A = \frac{1}{2}xy \). Differentiate this area formula with respect to time \( t \) to find \( \frac{dA}{dt} \): \( \frac{dA}{dt} = \frac{1}{2}(x \frac{dy}{dt} + y \frac{dx}{dt}) \).
Substitute the known values \( x = 12 \) ft, \( \frac{dx}{dt} = 5 \) ft/sec, and the calculated \( \frac{dy}{dt} \) into the differentiated area formula to find the rate at which the area of the triangle is changing.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
6mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Related Rates
Related rates are a method in calculus used to find the rate at which one quantity changes in relation to another. In this problem, we need to determine how the area of the triangle formed by the ladder, wall, and ground changes as the base of the ladder slides away from the wall. This involves differentiating the area with respect to time, using the known rates of change.
Recommended video:
04:16
Intro To Related Rates
Area of a Triangle
The area of a triangle can be calculated using the formula A = 1/2 * base * height. In this scenario, the base is the distance from the wall (x) and the height is the vertical distance from the ground to the top of the ladder (y). As the ladder slides, both x and y change, affecting the area, which we need to express in terms of these variables.
Recommended video:
05:06Finding Area When Bounds Are Not Given
Pythagorean Theorem
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). In this case, the ladder acts as the hypotenuse, and we can use the theorem (x^2 + y^2 = 13^2) to relate the height and base of the triangle, which is essential for finding the rate of change of the area.
Recommended video:
06:11Fundamental Theorem of Calculus Part 1
4:16mWatch next
Master Intro To Related Rates with a bite sized video explanation from Patrick
Start learningRelated Videos
Related Practice