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Numerade Educator

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Problem 31 Medium Difficulty

The sides of an equilateral triangle are increasing at a rate of $ 10 cm/min. $ At what rate is the area of the triangle increasing when the sides are $ 30 cm $ long?

Answer

$259.8 cm^{2} / min$

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Video Transcript

So here we get an example of a related rates problem. So we have an equilateral triangle where the three sidelines are equal. So for an equilateral triangle we have 60°, 60° and 60° where all the angles are congruent. So we know that the area of a triangle is one half base times height. Let's call the length of one side of our equilateral triangle. S. So this part, if we split the triangle in half, This region would be s divided by two. Well this region here is as radical three divided by two. Since we have a special triangle with angles of 33, and 93s. And as a result, area equals 1/2 base which is S. And heights S. Radical three divided by two. And this would be equivalent to Radical three divide by four S. Squared. So this would be the area of our triangle. So we're given information that dS over with respect to time Is equivalent to 10 cm permanent. And we're also giving information that S. is 30 cm. So for related to rates we have to take the derivative of both sides with respect to time. We have to be careful here since we have to use chain rule. So this would be the result of differentiation. So we know that this is equivalent to radical three times two Times 10 times 30. So this is equivalent to 150 radical three centimetres squared permanence. And this gives our final answer