Final answer:
The student's problem involves calculating the end area through the use of algebraic equations, integral calculus, differential equations, and trigonometric functions to estimate the volume of dirt needed to create a slope.
Step-by-step explanation:
The student's question pertains to the calculation of the end area in order to create a constant slope using various mathematical methods. The provided information suggests breaking the curve into four sections: 0-10 s, 10-20 s, 20-40 s, and 40-70 s, to calculate the end area accurately.
To start, we calculate the bottom rectangle which is common to all pieces, by multiplying 165 m/s by 70 s, giving us an area of 11,550 m. We then estimate a triangle at the top and calculate the areas for each section: Section 1 = 225 m; Section 2 = 550 m; Section 3 = 1,450 m; Section 4 = 2,550 m. By adding them, we obtain a net displacement of 16,325 m.
Using the tangent line and the given slope of 1 m/s², we can then apply integral calculus to calculate the area under the curve, which represents the displacement. The slope of the straight line between two points allows trigonometric functions to be used when calculating the angle and, subsequently, the end area.
Lastly, one could set up a differential equation if the rate of change of the slope is provided and needs to be factored into the model. This process represents diverse mathematical approaches including algebraic equations, calculus, and trigonometry for solving the real-world problem of estimating the volume of dirt needed to create a slope.