A particle is moving along the curve below.
y = sqroot(x)
As the particle passes through the point (16, 4), its x-coordinate increases at a rate of 4 cm/s. How fast is the distance from the particle to the origin changing at this instant? (Round your answer to three decimal places.)
y = x^0.5
dy /dx = 0.5 x^-0.5
dy/dt = (0.5 /sqrtx) dx/dt
if x = 16
dy/dt = ( 0.5 / 4 ) * 4 cm/s = 0.5
v = sqrt (dx/dt^2 + dy/dt^2)
= sqrt (4^2 + .25) = sqrt(16.25)
the distance z can be found using
z^2 = x^2 + y^2 = x^2 + x
2z dz/dt = (2x+1) dx/dt
so at (16,4)
2√260 dz/dt = (2*16+1)(4)
dz/dt = 132/(2√260) = 33/√65
To find how fast the distance from the particle to the origin is changing at the instant it passes through the point (16, 4), we need to use the concept of related rates.
The distance from the particle to the origin can be determined using the distance formula:
Distance = √(x^2 + y^2)
We are given that the particle is moving along the curve y = √(x), and its x-coordinate is increasing at a rate of 4 cm/s. To find the rate of change of distance from the origin, we need to differentiate the distance formula with respect to time (t) using the chain rule.
Let's start by differentiating the distance equation with respect to time:
d(Distance)/dt = d/dt [√(x^2 + y^2)]
Next, we can substitute the given equation y = √(x) to express the distance in terms of x:
d(Distance)/dt = d/dt [√(x^2 + (√(x))^2)]
Simplifying the expression:
d(Distance)/dt = d/dt [√(x^2 + x)]
Now, we can apply the chain rule:
d(Distance)/dt = (1/2) * (2x + 1) * dx/dt
We are given that dx/dt = 4 cm/s, and we need to find d(Distance)/dt when x = 16. So let's substitute the given values into the equation:
d(Distance)/dt = (1/2) * (2(16) + 1) * 4
Simplifying:
d(Distance)/dt = (1/2) * (32 + 1) * 4
= (1/2) * 33 * 4
= 66 cm/s
Therefore, the distance from the particle to the origin is changing at a rate of 66 cm/s at the instant it passes through the point (16, 4).
To find how fast the distance from the particle to the origin is changing at the point (16, 4), we need to use the formula for the distance between two points in a coordinate plane, which is given by the distance formula:
d = square root of ((x2 - x1)^2 + (y2 - y1)^2)
In this case, the point (x1, y1) is the origin (0, 0), and the point (x2, y2) is (16, 4).
Substituting these values into the distance formula, we get:
d = square root of ((16 - 0)^2 + (4 - 0)^2)
= square root of (16^2 + 4^2)
= square root of (256 + 16)
= square root of 272
= 16.492 cm (rounded to three decimal places)
We are given that the particle's x-coordinate is increasing at a rate of 4 cm/s. We want to find how fast the distance is changing, which means finding the derivative of the distance with respect to time.
Let's assume that t represents time.
So, d(t) = square root of ((x(t) - 0)^2 + (y(t) - 0)^2)
To find the derivative of d(t) with respect to time, we can differentiate both sides of the equation using the chain rule.
d'(t) = (1/2) * (x(t) - 0)^(-1/2) * (2*x'(t)) + (1/2) * (y(t) - 0)^(-1/2) * (2*y'(t))
Since we are interested in finding the derivative when x = 16, we can substitute this value into the equation and solve for d'(t) when x' = 4.
d'(t) = (1/2) * (16^(-1/2)) * (2*4) + (1/2) * (4^(-1/2)) * (0)
Simplifying this expression, we get:
d'(t) = (1/2) * (1/4) * 8
= 1 cm/s
Therefore, the distance from the particle to the origin is changing at a rate of 1 cm/s at the instant when the particle passes through the point (16, 4).