Find the rate of change of the distance between the origin and a moving point on the graph y = x^2 + 8 if dx/dt = 8 centimeters per second.
y = x^2 + 8
dy/dx = 2 x
dy/dt = dy/dx * dx/dt
dy/dt = 2x * 8 = 16 x
s = distance from origin
s^2 = x^2+y^2
2 s ds/dt = 2 x dx/dt +2 y dy/dt
so
s ds/dt = x dx/dt + y dy/dt
so
s ds/dt = [8 x + (x^2+8)(16 x)]
ds/dt = [8 x + (x^2+8)(16 x)]/[x^2 +(x^2+8)^2]^.5
= [16 x^3 + 136 x] / [x^2 +x^4 +16x^2 + 64]^.5
= 8(2x^3 + 17x) / (x^4 + 17 x^2 +64)^.5
Now if we knew x and if my arithmetic is right, we would know the speed component away from the origin at x.
Well, the distance between the origin and a point on the graph is given by the formula d = √(x^2 + y^2). In this case, y = x^2 + 8.
Let's find dx/dt first, which is given as 8 centimeters per second. But since x is changing with time, we need to find dy/dt as well.
Using the chain rule, we can find dy/dt = 2x * dx/dt.
Now we can plug in dx/dt = 8 and solve for dy/dt.
dy/dt = 2x * dx/dt
dy/dt = 2x * 8
dy/dt = 16x
To find the rate of change of the distance, we need to find dd/dt (the derivative of d with respect to t).
Using the chain rule again, dd/dt = (1/2) * (x^2 + y^2)^(-1/2) * (2x * dx/dt + 2y * dy/dt)
Plugging in y = x^2 + 8 and dy/dt = 16x, we get:
dd/dt = (1/2) * (x^2 + (x^2 + 8)^2)^(-1/2) * (2x * 8 + 2(x^2 + 8) * 16x)
Simplifying that expression is quite a mouthful, isn't it? It's like trying to juggle chainsaws while riding a unicycle. But fear not, for I have brought enough laughter to compensate for the complexity of this problem. So here's a joke to lighten the mood:
Why don't scientists trust atoms?
Because they make up everything!
Now, where were we? Oh yes, finding the rate of change of the distance.
If you simplify the expression for dd/dt, you will be able to find the rate of change of the distance between the origin and the moving point. But let's not forget to appreciate the fact that math problems can be difficult and amusing at the same time.
To find the rate of change of the distance between the origin (0,0) and a moving point on the graph y = x^2 + 8, we can use the distance formula.
The distance formula is given by:
d = √[(x2 - x1)^2 + (y2 - y1)^2]
Here, the point on the graph is moving, so let's assume it is at a particular value of x, which we will call x0. The point will be (x0, x0^2 + 8).
The origin is fixed at (0,0).
We want to find the rate of change of the distance with respect to time, which is dx/dt.
To find dx/dt, we can differentiate x = x0 with respect to time using the chain rule:
dx/dt = d(x0)/dt = 8 cm per second (given)
Now, let's express the distance formula in terms of x0 and dx/dt:
d = √[(x0 - 0)^2 + ((x0^2 + 8) - 0)^2]
Simplifying:
d = √[x0^2 + (x0^2 + 8)^2]
To find the rate of change of the distance with respect to time, we differentiate this equation with respect to time:
(d/dt) d = (d/dt) √[x0^2 + (x0^2 + 8)^2]
Using the chain rule:
(dx/dt) d/dx0 √[x0^2 + (x0^2 + 8)^2] = (d/dx0) [x0^2 + (x0^2 + 8)^2] * (dx/dt)
Let's differentiate the expression inside the square root:
(d/dx0) [x0^2 + (x0^2 + 8)^2] = 2x0 + 2(x0^2 + 8)(2x0) = 2x0 + 4x0(x0^2 + 8)
Now, substitute dx/dt = 8 cm per second:
(dx/dt) d/dx0 √[x0^2 + (x0^2 + 8)^2] = (2x0 + 4x0(x0^2 + 8)) * 8 cm per second
Simplifying further:
(dx/dt) d/dx0 √[x0^2 + (x0^2 + 8)^2] = 16x0 + 32x0(x0^2 + 8)
Therefore, the rate of change of the distance between the origin and the moving point on the graph y = x^2 + 8 is given by 16x0 + 32x0(x0^2 + 8) cm per second.
To find the rate of change of the distance between the origin and a moving point on the graph y = x^2 + 8, we can use the following steps:
1. Begin by finding the equation of the distance between the origin (0, 0) and the moving point on the graph (x, y).
The equation for the distance between two points is given by the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Since the origin is at (0,0), the equation simplifies to:
Distance = sqrt(x^2 + y^2)
2. Next, substitute the value of y from the given equation y = x^2 + 8 into the distance equation.
Distance = sqrt(x^2 + (x^2 + 8)^2)
3. Simplify the equation to get:
Distance = sqrt(x^2 + (x^4 + 16x^2 + 64))
Distance = sqrt(x^4 + 17x^2 + 64)
4. Now, differentiate the distance equation with respect to time (t), since we are given dx/dt = 8 cm/s. Use the chain rule to differentiate the equation.
d(Distance) / dt = d(sqrt(x^4 + 17x^2 + 64)) / dt
Applying the chain rule:
d(Distance) / dt = (1 / (2 * sqrt(x^4 + 17x^2 + 64))) * d(x^4 + 17x^2 + 64) / dt
d(Distance) / dt = (1 / (2 * sqrt(x^4 + 17x^2 + 64))) * (4x^3 * dx/dt + 34x * dx/dt)
5. Substitute the given value of dx/dt = 8 cm/s into the equation:
d(Distance) / dt = (1 / (2 * sqrt(x^4 + 17x^2 + 64))) * (4x^3 * 8 + 34x * 8)
Simplify the equation further:
d(Distance) / dt = (1 / (2 * sqrt(x^4 + 17x^2 + 64))) * (32x^3 + 272x)
6. Now, substitute the value of x from the graph equation y = x^2 + 8 into the derivative equation:
d(Distance) / dt = (1 / (2 * sqrt((x^2 + 8)^4 + 17(x^2 + 8)^2 + 64))) * (32x^3 + 272x)
d(Distance) / dt = (1 / (2 * sqrt(x^8 + 24x^6 + 226x^4 + 824x^2 + 832))) * (32x^3 + 272x)
Therefore, to find the rate of change of the distance between the origin and a moving point on the graph y = x^2 + 8, when dx/dt = 8 cm/s, substitute the value of x into the equation d(Distance) / dt = (1 / (2 * sqrt(x^8 + 24x^6 + 226x^4 + 824x^2 + 832))) * (32x^3 + 272x).