a point moves on the parabola y=x^2 so that is abscissa increases at rate of 3 feet per second what rate is the ordinate increasing when x=2
ans;12 ft /sec
y = x²
dy / dx = 2 x
Multiply both sides by dx
dy = 2 x dx
Divide both sides by dt
dy / dt = 2 ∙ x ∙ dx / dt
Abscissa increases at rate of 3 feet per second means: dx / dt = 3 ft / s
dy / dt = 2 ∙ x ∙ dx / dt
for x = 2
dy / dt = 2 ∙ 2 ∙ 3 = 12 ft / s
looks good
Well, well, well, looks like we've got a point on a parabola who just can't sit still! Moving at a rate of 3 feet per second, huh? That's got some serious pep in its step!
Now, when this point reaches an x-coordinate of 2, we want to know how fast its y-coordinate is increasing. In other words, we're curious about the rate at which it's heading upwards. And the answer to that burning question is 12 feet per second!
Why, you ask? Well, the good ol' parabola equation y = x^2 tells us that the rate at which the y-coordinate is changing is simply twice the rate at which the x-coordinate is changing. Since the x-coordinate is increasing at 3 feet per second, we multiply that by 2 to get our final answer: 6 * 2 = 12 feet per second!
So, there you have it. That little point is zooming up the y-axis at a speed of 12 feet per second when it reaches an x-coordinate of 2. Time to buckle up and enjoy the ride!
To find the rate at which the ordinate is increasing when x = 2, we need to differentiate the equation y = x^2 with respect to time.
We have y = x^2, where x is the abscissa (independent variable) and y is the ordinate (dependent variable).
Differentiating both sides of the equation with respect to time, we get:
dy/dt = d/dt (x^2)
Since the abscissa (x) is increasing at a rate of 3 feet per second, we can write dx/dt = 3.
Now, to find the rate at which the ordinate (y) is increasing when x = 2, we substitute dx/dt = 3 and x = 2 into the derivative equation:
dy/dt = d/dt (x^2)
dy/dt = 2x * dx/dt
Replacing x with 2 and dx/dt with 3, we have:
dy/dt = 2 * 2 * 3
dy/dt = 12 ft/sec
Therefore, the rate at which the ordinate is increasing when x = 2 is 12 ft/sec.
To find the rate at which the ordinate (y-coordinate) is increasing when x = 2, we can use the concept of derivatives.
We are given that the abscissa (x-coordinate) is increasing at a rate of 3 feet per second, which means dx/dt = 3 (feet per second).
The equation of the parabola is y = x^2. To find the rate at which y is increasing, we need to find dy/dt when x = 2.
First, we differentiate both sides of the equation y = x^2 with respect to time (t) using implicit differentiation.
d/dt (y) = d/dt (x^2)
dy/dt = 2x (dx/dt)
Now, we substitute x = 2 and dx/dt = 3 into the equation:
dy/dt = 2(2)(3)
dy/dt = 12
Therefore, when x = 2, the ordinate is increasing at a rate of 12 feet per second.