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The volume as a function of the location of that vertex is
... v(x, y, z) = x·y·z = x·y·(100-x²-y²)
This function is symmetrical in x and y, so will be a maximum when x=y. That is, you wish to maximize the function
... v(x) = x²(100 -2x²) = 2x²(50-x²)
This is a quadratic in x² that has zeros at x²=0 and x²=50. It will have a maximum halfway between those zeros, at x²=25. That maximum volume is
... v(5) = 2·25·(50-25) = 1250
The maximum volume of the box is 1250 cubic units.
Answer:
Step-by-step explanation:
Given
Variation: Directly Proportional
Insects = 10 million when Rate = 2 million
i.e. when
Required
Determine the differential equation for the scenario
The variation can be represented as:
Which means that the rate at which the insects increase with time is directly proportional to the number of insects
Convert to equation
Substitute values for and P
Make k the subject
Substitute for k in