a particle is moving along the graph of y=x^(1/3) when x=8 the y component of its position is increasing at the rate of 1 centimeter per second how fast is the x component changing at this moment
y = x^(1/3)
dy/dt = 1/3 ÷ x^(2/3) dx/dt
plug in the numbers:
1 = 1/3 ÷ 4 dx/dt
12 = dx/dt
Well, well, well! Looks like we have a particle on a math adventure, eh? Let's have some fun with this one!
First of all, we know that the particle is moving along the graph of y = x^(1/3). So, we can use this information to find the relationship between x and y.
Now, we have the x-component changing (let's call it dx/dt) and the y-component changing (let's call it dy/dt).
Given that the y-component is increasing at a rate of 1 centimeter per second, we can say that dy/dt = 1 cm/s.
To find how fast the x-component is changing at this moment, we need to find dx/dt.
Are you ready for the punchline? Since we have the equation y = x^(1/3), we can differentiate it with respect to time (t) to find the relationship between dx/dt and dy/dt.
Ready for it? Drumroll, please...
dy/dt = (1/3) * (x^(-2/3)) * dx/dt
Now, since dy/dt = 1 cm/s, we can plug in the values and solve for dx/dt. Are you bracing yourself for the answer?
1 cm/s = (1/3) * (8^(-2/3)) * dx/dt
Now, let's simplify it like a true math clown:
1 cm/s = (1/3) * (1/2) * dx/dt
1 cm/s = (1/6) * dx/dt
And now, to find the true speed of the x-component change:
dx/dt = 6 cm/s
So, the x-component is changing at a rate of 6 centimeters per second. Voilà! Clown math at its finest.
To find how fast the x-component is changing at the moment when x = 8, we can use the chain rule of differentiation.
Given:
dy/dt = 1 cm/s (the rate at which the y-component is changing)
x = 8
We need to find dx/dt (how fast the x-component is changing).
First, let's differentiate both sides of the equation y = x^(1/3) with respect to time (t):
d/dt (y) = d/dt (x^(1/3)).
Using the chain rule, we can rewrite this as:
dy/dx * dx/dt = (1/3) * x^(-2/3) * dx/dt.
Rearranging the equation, we have:
dx/dt = (3 * dy/dx * x^(2/3)).
Now, let's substitute the given values:
dy/dx = 1 (as the rate of change of y with respect to x is not given in the problem)
x = 8
Plugging in these values, we get:
dx/dt = 3 * 1 * 8^(2/3).
Simplifying this expression, we have:
dx/dt = 3 * 8^(2/3) cm/s.
Calculating 8^(2/3), we get:
8^(2/3) = (8^2)^(1/3) = 64^(1/3) = 4.
Therefore, the x-component is changing at a rate of:
dx/dt = 3 * 4 cm/s = 12 cm/s.
To determine how fast the x component is changing at a given moment, we can use the chain rule from calculus. The chain rule states that if y is a function of x, and x is a function of t (time), then the rate of change of y with respect to t can be calculated by multiplying the rate of change of y with respect to x by the rate of change of x with respect to t.
In this case, we are given that y = x^(1/3), and we want to find the rate of change of x with respect to t when x = 8. To do this, we first need to express x as a function of t. However, since we only have the equation for y in terms of x, we need to solve for x in terms of y.
Raising both sides of the equation y = x^(1/3) to the power of 3, we get:
y^3 = x^(1/3)^3
y^3 = x
Now, we have x in terms of y: x = y^3.
Since we know that the y component of the position is increasing at a rate of 1 centimeter per second, we can say that dy/dt = 1.
To find dx/dt, the rate at which x is changing with respect to t, we can differentiate both sides of the equation x = y^3 with respect to t using implicit differentiation:
d/dt(x) = d/dt(y^3)
Now, we apply the chain rule to differentiate y^3 with respect to t:
dx/dt = 3y^2 * dy/dt
Substituting the values we know, with y = (8)^(1/3) because it is given that x = 8:
dx/dt = 3(8)^(2/3) * 1
Simplifying, we have:
dx/dt = 3 * (2^2) * (2^(1/3))
dx/dt = 3 * 4 * 2^(1/3)
dx/dt = 12 * 2^(1/3)
Therefore, the x component is changing at a rate of 12 * 2^(1/3) centimeters per second when x = 8.