Next year, if Company Q produces x units of a certain product and sells y units of this product,...

Question

Next year, if Company Q produces x units of a certain product and sells y units of this product, its total profit, in dollars, from this product will be 10y - 4x. If x and y are related by the equation y = rx and if Company Q's average (arithmetic mean) profit per item produced next year must be at least $4.50, what is the least possible value of r?

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E We translate "average profit per item produced next year must be at least 4.50" into an inequality comparing profit per item to 4.50. 10y - 4x/x 4.50 We substitute "y = rx" into the inequality to express everything in terms of r and x. 10(rx) - 4x/x 4.50 We simplify the numerator by factoring out x and then cancel x from numerator and denominator. x(10r - 4)/x 4.50 10r - 4 4.50 We add 4 to both sides of the inequality to isolate the term with r. 10r 4.50 + 4 We divide both sides by 10 to solve for r. r 8.50/10 = 0.85

Explanation	Calculations	Help
$We translate "average profit per item produced next year must be at least 4.50" into an inequality comparing profit per item to 4.50.$	$\frac{10 y - 4 x}{x} \geq 4.50$
$We substitute "y = rx" into the inequality to express everything in terms of r and x.$	$\frac{10 ( r x ) - 4 x}{x} \geq 4.50$
$We simplify the numerator by factoring out x and then cancel x from numerator and denominator.$	$\frac{x ( 10 r - 4 )}{x} \geq 4.50 10 r - 4 \geq 4.50$
$We add 4 to both sides of the inequality to isolate the term with r.$	$10 r \geq 4.50 + 4$
$We divide both sides by 10 to solve for r.$	$r \geq \frac{8.50}{10} = 0.85$

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