Strength s = kbd^3 for some constant k, and the perimeter condition enables you to express d in terms of b. So you have s as a function of b, and you are looking for the value(s) of b that maximises s...
Do you know what to do?
Hey i have a question im getting no where with:
The strength of a rectangular beam of given length is proportional to bd^{3}where b is the breadth and d the depth. If the cross section of the beam has a perimeter of 4m, find the breadth and depth of the strongest beam.
Thanks.
So the context isn't exercises in finding maximum and minimum points (turning points) on the curve of a function? (By finding where the derivative = 0 ?) You haven't been doing exercises of that sort?
Do you mean you can't get d in terms of b? But the length of the perimeter is 4 = 2b + 2d...
i got that as the perimeter its just i was told by a friend that this has something to do with parametric equations, and yes i have been doing exercises of that sort. But in my revision notes I have no worked examples of a problem like this hence why im stuck and i have a mid term for this module next week.
Ok, it could be an exercise in Lagrange multiplier - Wikipedia, the free encyclopedia.
That way you don't eliminate b or d at the beginning, but find critical points of the two-variable function s = kbd^3, given the constraint, 2b + 2d = 4.
Same thing in the end, of course. You could try plugging the function and constraint straight into the formula (on the wiki page), but your best chance of understanding is to revise the basics by optimising a function of b as I suggested.
I.e. begin by making d the subject of 2b + 2d = 4. Get s in terms of k and b. Differentiate with respect to b, set the derivative equal to zero. That's where the gradient of the curve is zero and s is therefore a max or min.
you sure its about lagrange multiplier as thats not on our topic list, the topic list is
Differentiation of trig and log functions, product, quotient and function of function rules. Higher derivatives.
Hyperbolic and inverse hyperbolic functions and derivatives. Log equivalents of hyperbolic functions. Differentiation of inverse hyperbolic and inverse trig functions.
Applications of differentiation:Tangents speed and acceleration. Rates of Change. Local maxima and minima and applications
implicit and parametric differentiation. Logarithmic differentiation
partial differentiation