Decide whether the following problem can be solved using precalculus, or whether calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, explain your reasoning and use a graphical or numerical approach to estimate the solution.

Find the distance traveled in 45 seconds by an object traveling at a velocity of v(t) = 20 + 3cos(t) feet per second. (Round your answer to the nearest foot.)

distance=INTEGRAL velocity*dtime

In cos t I assume t is in degrees. If you mean radians, then do it that way.

It is a calculus problem because every little bit of distance is v(t) times a tiny bit of time, delta t. It v(t) were constant you could just multiply v times change in time to get the answer, but v is not constant.
A primitive way to do this is to split the 45 seconds up into three 15 second periods. Approximate that it goes v(0) for the first 15, then v(15) for the second 15 (15 to 30), then v(30 for the last 15 (30 to 45)
v(0)(15) = 23 * 15 = 345
v(15) * 15 = 22.89 * 15 = 343.5
v(30) * 15 = 22.6 * 15 = 339
so
d = 1027.5
Next divide it into 5 spaces of 9 seconds each and see how much the answer changes.

I suspect that is not allowed Bob :) Too easy.

Find the distance traveled in 15 seconds by an object traveling

at a constant velocity of 20 feet per second.

To determine whether this problem can be solved using precalculus or if calculus is required, we need to analyze the given information.

The problem provides the velocity function v(t), and we want to find the distance traveled in 45 seconds. Recall that distance is the accumulated total of the absolute values of the velocities over the given time interval. In other words, we need to find the definite integral of the velocity function over the interval [0, 45].

Since the velocity function v(t) involves a trigonometric function (cosine), it is not a basic polynomial or rational function. Calculus is required to evaluate the definite integral of this function.

To estimate the solution using a graphical or numerical approach, we can plot the graph of the velocity function v(t) and use a numerical method to approximate the area under the curve. One such method is the midpoint rule, which divides the interval into equal subintervals and calculates the sum of the areas of rectangles with heights equal to the function values at the midpoints of each subinterval.

Here's how you can estimate the solution using the midpoint rule:

1. Plot the graph of the velocity function v(t):
- Choose a suitable range for t, such as [0, 45].
- Create a graph with the t-axis as the horizontal axis and the v(t)-axis as the vertical axis.
- Use a graphing calculator or software to plot the graph of v(t) = 20 + 3cos(t).

2. Divide the interval [0, 45] into equal subintervals:
- Determine the number of subintervals you want to use for the approximation. The larger the number, the more accurate the estimate will be. Let's say we use n subintervals.

3. Find the midpoint of each subinterval:
- Calculate the width of each subinterval by dividing the total interval width (45 - 0 = 45) by the number of subintervals: Δt = 45/n.
- The midpoints of the subintervals can be calculated as: t1 = Δt/2, t2 = Δt + Δt/2, t3 = 2Δt + Δt/2, and so on, up to tn = nΔt + Δt/2.

4. Evaluate the velocity function at each midpoint:
- Plug in the values of each midpoint (t1, t2, t3, ..., tn) into the velocity function v(t) = 20 + 3cos(t) to find the corresponding velocity values (v1, v2, v3, ..., vn).

5. Calculate the sum of the rectangle areas:
- Multiply each velocity value by the width of the subinterval (Δt) to get the area of each rectangle: A1 = v1 * Δt, A2 = v2 * Δt, A3 = v3 * Δt, ..., An = vn * Δt.
- Calculate the total area by summing up all the rectangle areas: Total area ≈ A1 + A2 + A3 + ... + An.

The result of this calculation will give you an estimate of the distance traveled by the object in 45 seconds. Remember to round your answer to the nearest foot, as requested in the problem.

Note: While calculus is required to find the exact solution using integration, the numerical approximation provided by the midpoint rule should give you a reasonably close estimate.