In a right triangle, the hypotenuse is of fixed length of 15 units, one side is increasing in length by 4 units per second while the third side is decreasing in size. At a certain instant the increasing side is of length 9 units. Find the rate of change of the third side at the same instant.
Multiple choice answers
a. -3 units per second
b. -5/2 units per second
c. -3/2 units per second
d. -5/4 units per second
e. -3/4 units per second
which on is it? I got -51/8 units per second but that's not right
we have
x^2+y^2 = 15^2
x dx/dt + y dy/dt = 0
Let x be decreasing, and y increasing. Then when y=9, x=12, and we have
12 dx/dt + 9*4 = 0
dx/dt = -3
To find the rate of change of the third side of the right triangle, we need to use the Pythagorean theorem. Let's denote the sides of the triangle as follows:
- Hypotenuse: c = 15 units (fixed length)
- Increasing side: a (increasing at a rate of 4 units per second)
- Third side (decreasing): b
Using the Pythagorean theorem, we have the equation:
a^2 + b^2 = c^2
Now, let's differentiate both sides of the equation with respect to time:
2a(da/dt) + 2b(db/dt) = 0
Since we are interested in finding the rate of change of the third side (db/dt) when the increasing side is 9 units, we can substitute the given values into the equation:
2(9)(4) + 2b(db/dt) = 0
72 + 2b(db/dt) = 0
Simplifying the equation, we have:
2b(db/dt) = -72
Dividing both sides by 2b, we get:
db/dt = -72 / (2b)
Now, let's substitute the value of the increasing side (a) at that instant. We know that at that instant, a = 9 units. Since it is a right triangle, we can use the Pythagorean theorem to find the value of b:
9^2 + b^2 = 15^2
81 + b^2 = 225
b^2 = 225 - 81
b^2 = 144
b = 12 units
Substituting the values of b = 12 units and a = 9 units into the equation for db/dt, we get:
db/dt = -72 / (2 * 12)
db/dt = -72 / 24
Simplifying further, we have:
db/dt = -3 units per second
Therefore, the correct answer is (a) -3 units per second.
To solve this problem, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Let's denote the increasing side as x, the decreasing side as y, and the hypotenuse as 15. We are given that dx/dt (the rate of change of x) is 4 units per second.
Using the Pythagorean theorem, we have the equation:
x^2 + y^2 = 15^2
Differentiating both sides of the equation with respect to time t, we get:
2x(dx/dt) + 2y(dy/dt) = 0
Now, we substitute x = 9 (since at the given instant, x = 9 units) and solve for dy/dt:
2(9)(4) + 2y(dy/dt) = 0
72 + 2y(dy/dt) = 0
Divide both sides by 2y:
(dy/dt) = -72 / (2y)
Now, we need to find the value of y. Since the triangles are similar, we can set up the following proportion:
x / y = 15 / 9
Simplifying this equation, we get:
x = (5/3)y
Substituting the value of x = 9, we can solve for y:
9 = (5/3)y
27 = 5y
y = 27/5
Now, substitute this value of y into the dy/dt equation to find the rate of change of the third side:
(dy/dt) = -72 / (2 * 27/5)
(dy/dt) = -72 / (54/5)
(dy/dt) = -72 * (5/54)
(dy/dt) = -360/54
(dy/dt) = -20/3
(dy/dt) = -6.67 units per second (rounded to two decimal places)
However, none of the multiple-choice options matches this value exactly. So, it's possible that there was an error made in the calculations or equation setup. It's recommended to recheck the solution steps and calculations to identify any potential mistakes.