The circumference of the circular robotic path with a radius of 10 meters is 63 meters when rounded to the nearest whole number. The slope of the tangent line at the origin (0,0) is undefined.
To calculate the circumference of the robotic path, we use the formula for the circumference of a circle, which is C = 2πr. Since the given radius (r) is 10 meters, we have:
C = 2 × 3.14 × 10 = 62.8 meters.
Rounded to the nearest whole number, the circumference is 63 meters.
The equation of the circle given is x^2 + y^2 = 100. To find the slope of the tangent line at a point on the circle, we can differentiate this equation implicitly with respect to x:
2x + 2y(dy/dx) = 0
Solving for dy/dx (the slope), we get:
dy/dx = -x/y
At the starting point (0,0), the slope of the tangent line cannot be determined as it becomes a form of 0/0 which is indefinite. Therefore, the slope of the tangent line at the origin is undefined.
The probable question may be:
In a technological context, a circular robotic path with a radius of 10 meters needs its circumference calculated. Utilizing the constant value of 3.14 for π and rounding to the nearest whole number, determine the circumference of the robot's circular trajectory. Additionally, provide the slope of a tangent line to the circle at a specific point, rounding the value to three decimal places. Consider the point where the robot starts its motion as the origin (0,0) and provide the coordinates of this point. Ensure you use the appropriate variables in your calculations, and don't forget to use the Preview button to confirm your syntax before submitting your answer.
Additional Information:
The robotic path is represented by the equation of the circle: x^2+y^2=100.
The robot starts its motion from the point (0,0).
The slope of the tangent line is determined at the starting point of the robot's trajectory.
The units for all measurements are in meters