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Problem 78 Hard Difficulty

Find the value of $ c $ such that the line $ y = \frac{3}{2}x + 6 $ is tangent to the curve $ y = c \sqrt{x}.

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Video Transcript

it's lower. So when you read here So we're gonna start off by differentiating C square affects when we get C over two square root of X. This is for the line. Why equals three Half plus six is tangent to the curve, See skirt of X. So the slope off the line three over two should be equal to the derivative. So we see that this is equal to three over two. When we get our X value to be C square over nine, then we're going to solve the equation. So we get why is equal to three Have X Plus six we have. Why is equal to see square it of X. Don't we get three has X Plus six is equal to C square of X. Then we're going to simplify the equation by squaring both sides when we get nine over four Up Square plus 18 x was 36 is equal to see square necks, and we're going to simplify our equation even more. We got 9/4 c square over nine, which we got right here square plus 18 c square over nine plus 36 this equal to see square times C square over nine, and then we just simplify continuously. You see that? It's only positive. Um, Val, You see, since the only positive value sea gives a curve that intersects the line.