Here is the question:

"Engineers have determined that the strength s of a rectangular beam varies as the product of the width w and the square of the depth d of the beam; that is, s = kwd^2 for some constant k. Find the dimensions of the strongest rectangular beam that can be cut from a cylindrical log with diameter 48cm."

There is a drawing (which I don't have the technology or know how to present for you); it shows the rectangle inside the circle with the diagonal of the rectangle (which is the diameter of the circle) at 48 cm and the lines/sides to the right angular corner of the rectangle on the circumference, being labelled w(width) and d (depth)

My reasoning so far:

Relevant is Pythagoras' Theorem, making w^2 +d^2 =48^2 and making d^2 the subject d^2=(48^2-w^2), and differentiating the equation for strength ds/dd = 2kwd

Make this equal to zero to find the maximum

I am having difficulty in knowing how to progress with this so w and d can be isolated. I am tempted to substitute d^2 with (48^2 - w^2) into the equation for s, but it doesn't help isolate the terms because there is still the constant k involved.

"Engineers have determined that the strength s of a rectangular beam varies as the product of the width w and the square of the depth d of the beam; that is, s = kwd^2 for some constant k. Find the dimensions of the strongest rectangular beam that can be cut from a cylindrical log with diameter 48cm."

There is a drawing (which I don't have the technology or know how to present for you); it shows the rectangle inside the circle with the diagonal of the rectangle (which is the diameter of the circle) at 48 cm and the lines/sides to the right angular corner of the rectangle on the circumference, being labelled w(width) and d (depth)

My reasoning so far:

Relevant is Pythagoras' Theorem, making w^2 +d^2 =48^2 and making d^2 the subject d^2=(48^2-w^2), and differentiating the equation for strength ds/dd = 2kwd

Make this equal to zero to find the maximum

I am having difficulty in knowing how to progress with this so w and d can be isolated. I am tempted to substitute d^2 with (48^2 - w^2) into the equation for s, but it doesn't help isolate the terms because there is still the constant k involved.

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