Related Rates and a Trapezoidal Trough
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- čas přidán 29. 08. 2024
- In this video, we solve a related rates problem involving a filling trough of water. It involves implicit differentiation of the volume formula of a trapezoidal prism.
This is lecture 29 (part 2/4) of the lecture series offered by Dr. Andrew Misseldine for the course Math 1210 - Calculus I at Southern Utah University. A transcript of this lecture can be found at Dr. Misseldine's website or through his Google Drive at: drive.google.c...
This lecture is based upon Section 3.9 of Calculus by James Stewart. Please post any questions you might have below in the comment field and Dr. Misseldine (or other commenters) can answer them for you. Please also subscribe for further updates.
I've been stuck on this problem for a long time, simply because I've been working through this problem by converting Meters to Centimeters. I'm really desperate to understand why the unit conversion here matters so much. Is there something I'm overlooking?
You want all of your units to be the same with respect to length so that you're comparing apples to apples. given that the change of volume with respect to Time was measured in cubic meters per minute, we do ourselves a big favor by converting all of the lengths in the meters. Admittedly go the other way around switch the meters to centimeters, I simply thought that's switching all the centimeters to meters will be the simplest of the conversions. On the other hand if you allow meters and centimeters to coexist your final answer is going to be off by a power of 10. Each conversion from a centimeter to a meter will cause you to divide by a hundred. As ultimately we have to measure volume you have to convert centimeters into meters three times changing you by a fact of 100 cubed.
@@Misseldine I'm sorry I must clarify. In the beginning I've converted all of my measurements to centimeters, but even though I've pretty much done every step, I still can't get the right answer :/
Is the error off by only a decimal place?
@@Misseldine I checked in with the teacher and the mistake was the unit conversion. I was supposed to convert 100m^3 to 1,000,000cm^3. Which I failed to recognize.
I think we skipped the formula for Area of the face here. Did u just go straight to plugging in the values?? 10h(.3 +a) ? and is "a" the whole horizontal line or just the line inside the triangle??
Around 4 minutes we derive a formula for the formula of the trapezoid. We explain it has the same area as a rectangle whose area is h(0.3+a). Recall that the 10 is the length of the whole trough and had nothing to do with the area of the trapezoid.
The quantity a, as labeled on the figure, is the distance from the slant of the trapezoid to the perpendicular connecting the end point of the bottom with the top. Hence, a is the width of a right triangle. This is why the width of the associated rectangle is 0.3+a.
isn't the area formula 1/2 (a+b)(h)? Why did you write it as (a+b)(h)
so, then the volume will be = 1/2(l)(h) (a+b)->(5)(h) (a+b) not 10(h)(a+b)
or did I do something wrong?
But translating the triangle on the left over to the triangle on the right the trapezoid is equivalent to a rectangle. As such the area of a rectangle is length times width.
hehehe
Glad you liked it.