Optimization practice question 12
A poster is to carry cm² of printed text, with margins of cm at the top and the bottom and cm at each side. Let be the width of the printed text, in centimetres. Write the total area of the poster, in cm², as a function of alone.
If the printed text is cm wide, how tall is it? Then add the margins to each dimension.
The text is cm tall. Two side margins widen the poster, and a top and a bottom margin make it taller.
A formula in x. Use ^ for powers and brackets for every input: sin(2x), e^(x^2), log(x+1). Give exact values: 1/6, not 0.167.
Theory UBC Math 100 §8.3CLP-1 §3.5Key results
Worked solution
Strategy
Express the text’s height through its area, add the margins to each dimension, and multiply. The constraint, 200 cm² of text, is what leaves x as the only variable.
Working
Its area is 200 cm².
A margin on each side, and one at the top and one at the bottom.
Width times height, expanded.
Answer
The answer is A equals x plus 4 times 200 over x plus 8, which is 8x plus 800 over x plus 232.
Check
With text 10 cm wide and 20 cm tall, the poster is 14 by 28, or 392 cm², and 8 · 10 + 800/10 + 232 = 392.
Where this goes wrong
Adding each margin once. There are two side margins and two end margins, so the width grows by 4 and the height by 8.
The idea behind it
The optimization lesson, a figure to move, in a few steps
Leans on
Writing a function from a word problem, from Functions, graphs and inverses
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