At a certain moment a race official is watching a race care approach the finish line along a straight track at some constant, positive speed. Suppose the official is sitting still at the finish line, 20m from the point where the car will cross. Question: at the time described what is the sign of d/dt (dx/dt)?positivenegative zero ### Diagram ExplanationThe diagram illustrates a right triangle formed by the path of a car and a viewer located at a distance from the finish line. Key elements in the diagram include:- **Finish Line**: Marked in blue on the right side.- **Car**: Positioned at the starting point on the left side, indicated moving towards the finish line.- **Right Triangle**: Formed by distances \(x\), \(y\), and hypotenuse \(L\). ### Variables- **\(x\)**: The horizontal distance from the car to a point along the finish line.- **\(y\)**: The vertical distance from that point on the finish line to the viewer.- **\(L\)**: The hypotenuse representing the direct line of sight from the viewer to the car.- **\(\theta\)**: The angle of elevation from the viewer's line of sight to the car.This arrangement can be used in mathematical problems involving trigonometry, specifically right triangle relationships. The viewer appears to be observing the car's movement, and the angle \(\theta\) could be calculated using trigonometric functions depending on the given values of \(x\), \(y\), or \(L\). The image is a mathematical expression showing a derivative operation. It displays the second derivative of a function with respect to time.Expression: \[\frac{d}{dt} \left( \frac{dx}{dt} \right)\]Explanation:- The expression represents the derivative of the first derivative of a function \( x \) with respect to time \( t \). - The inner part, \(\frac{dx}{dt}\), is the first derivative, indicating the rate of change of \( x \) with respect to time.- The entire expression indicates the second derivative, representing the acceleration, or the rate of change of the rate of change of \( x \) over time.

Name: At a certain moment a race official is watching a race care approach t
Uploaded: 2024-03-05T11:56:34.000Z
Channel: Bartleby
Description: At a certain moment a race official is watching a race care approach the finish line along a straight track at some constant, positive speed. Suppose the official is sitting still at the finish line, 20m from the point where the car will cross. Quest

Introduction: Observe the position of the car, the race official and finish line, they all form right angle . The distance between the point where the car crosses while moving in a straight line and the race official is 20 m. Therefore y=20 m. Now determine the value of trigonometric function tan of an angle theta. tanθ=yxSubstitute 20 m for y in above equation. tanθ=20 mxx=20 m tan θ Differentiate both sides with respect to t. dxdt=20 m 1tanθdθdt=20 m cotθdθdt=20 m -cosec2θdθdtTo calculate the value of ddtdxdt, differentiate both side with respect to t again. ddtdxdt=20 m ddt-cosec2θdθdt=20 m 2cosec2θ cot θ d2θdt2 =40 m cosec2θ cot θ d2θdt2 Therefore, at the given instant of sign of ddtdxdtis positive.