(Another) Introductory Level Calculus Question

• Wormaldson
In summary, the cross section of a canal can be modeled by a parabola with a vertex at (0, -20) and ends at (-40, 0) and (40, 0). Using the equation y = ax2 + bx + c, the volume of water in the canal when it is full to ground level can be found by taking the integral of the equation from -40 to 40 and multiplying by the length of the canal, 0.8 km.
Wormaldson

Homework Statement

The cross section of a canal can be modeled by a parabola.
The canal is 80 m wide at ground level and 20 m deep at its lowest point.
The canal is 0.8 km long.
Find the volume of water in the canal when it is full to ground level.

Homework Equations

Will add them once I find out what they actually are.

The Attempt at a Solution

I've deduced so far that the derivative of the parabola (if that's even relevant to the question) will cross the x-axis axis at the vertex/lowest point of the canal. This point is at 40 metres across.

Problem is, I don't know what to do with this information. I'm assuming I'm supposed to construct the equation of the parabola somehow, but I don't know how to do that.

Any help would be much appreciated.

its an integration problem

granpa said:
its an integration problem

Thanks, that's a step in the right direction.

Wormaldson said:

Homework Statement

The cross section of a canal can be modeled by a parabola.
The canal is 80 m wide at ground level and 20 m deep at its lowest point.
The canal is 0.8 km long.
Find the volume of water in the canal when it is full to ground level.

Homework Equations

Will add them once I find out what they actually are.

The Attempt at a Solution

I've deduced so far that the derivative of the parabola (if that's even relevant to the question) will cross the x-axis axis at the vertex/lowest point of the canal. This point is at 40 metres across.
You don't need the derivative to determine this. You are given that the cross-section shape is a parabola. Any parabola can be modeled by the equation y = ax2 + bx + c, for some constants a, b, and c.

Obviously, the parabola opens upward, so a > 0. You can put the vertex wherever you like, but it might be more convenient to put it at (0, -20), and the two ends at (-40, 0) and (40, 0). Use these three points to find the equation of your parabola.
Wormaldson said:
Problem is, I don't know what to do with this information. I'm assuming I'm supposed to construct the equation of the parabola somehow, but I don't know how to do that.

Any help would be much appreciated.

Wormaldson said:
Problem is, I don't know what to do with this information. I'm assuming I'm supposed to construct the equation of the parabola somehow, but I don't know how to do that.

Sounds like you need to do some review. You should know how to do this before getting into introductory calculus.

Mark44 said:
You don't need the derivative to determine this. You are given that the cross-section shape is a parabola. Any parabola can be modeled by the equation y = ax2 + bx + c, for some constants a, b, and c.

Obviously, the parabola opens upward, so a > 0. You can put the vertex wherever you like, but it might be more convenient to put it at (0, -20), and the two ends at (-40, 0) and (40, 0). Use these three points to find the equation of your parabola.

Thank you sir, I think I've got it now. Will update with the working when I get the time to do it.

zgozvrm said:
Sounds like you need to do some review. You should know how to do this before getting into introductory calculus.

I'm aware of that, but I don't have much of a choice, really. My exams are coming up fast, and I'm aiming to score well in them, so I have to sort of "force-feed" myself this stuff as quickly as I can.

Anyway, though, I worked out how to construct the equation for the parabola and find the integral very quickly after reading Mark's advice, so I think I'll be alright. The calculus paper in my exam is ultra-fundamental stuff, and the question I posted would be expected to be one of the most difficult questions. Hopefully that gives a better perspective of the difficulty of what I'm being tested on.

1. What is calculus?

Calculus is a branch of mathematics that deals with the study of continuous change. It involves the analysis of functions and their rates of change, known as derivatives, as well as the accumulation of quantities over intervals, known as integrals.

2. What is the purpose of learning calculus?

The purpose of learning calculus is to develop a deeper understanding of how things change and to be able to solve problems involving rates of change, optimization, and accumulation. It is also a necessary foundation for many other fields of study, such as physics, engineering, and economics.

3. What are the two main branches of calculus?

The two main branches of calculus are differential calculus, which deals with rates of change and slopes of curves, and integral calculus, which deals with the accumulation of quantities and the area under curves.

4. What is the difference between limits and derivatives?

Limits and derivatives are closely related concepts in calculus. A limit is the value that a function approaches as the input approaches a certain value, while a derivative is the instantaneous rate of change of a function at a specific point. In other words, a derivative is the slope of the tangent line to a curve at a given point, while a limit is the slope of the curve as the distance between two points becomes smaller and smaller.

5. What are some real-world applications of calculus?

Calculus has many real-world applications, such as in physics (e.g. calculating the velocity and acceleration of objects), engineering (e.g. designing structures and optimizing systems), economics (e.g. modeling supply and demand), and statistics (e.g. analyzing data and making predictions). It is also used in fields like medicine, computer science, and finance.

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