Question

At what value(s) of ___ on the curve does the tangent line have the largest slope?

A) x
B) y
C) z
D) t

Answer 1

To find where the tangent line to a curve has the largest slope, determine the curve's endpoint positions at different times and use the slope formula. The largest slope represents the highest velocity over the shortest time period.

Step-by-step explanation:

To determine at what value of t on the curve the tangent line has the largest slope, one must understand the concept of the slope of a tangent line to a curve. In this context, slope refers to the rate of change of a function, or how much y changes per unit change in t.

Based on the information provided, the steps to find the tangent line's slope at t = 25 s include:

1. The slope of a curve at a point is equal to the slope of a straight line tangent to the curve at that point. To find the tangent line with the largest slope on the curve, we need to find the point where the tangent line has the steepest slope.

1. We are given that the tangent line is to be drawn at t = 25 s and that it passes through the points (19, 1300) and (32, 3120). We can use these points to find the slope of the tangent line by using the formula:

1. Slope (m) = (change in y) / (change in x).

1. By substituting the values into the formula, we get:

1. m = (3120 - 1300) / (32 - 19) = 1820 / 13 ≈ 140.

The largest slope corresponds to the greatest change in position over the shortest amount of time, implying the highest velocity for a given time interval on the curve.