41. Among all triangles in the first quadrant formed by the x-...

Question

41. Among all triangles in the first quadrant formed by the x-axis, the y-axis, and tangent lines to the graph of y=3x-x^2, what is the smallest possible area?

Graph showing a blue curve of y=3x-x², a tangent line, and a shaded triangle in the first quadrant.

Accepted Answer

Welcome back, everyone. Find the area of the largest possible rectangle inscribed in the region bounded by quadrant 1, and the curve Y equals 4 minus X2. So we're given the sketch for this problem. Let's identify the area of any rectangle which meets those conditions. We know that area of a rectangle is simply found by multiplying length and the width. Now, in this context, we can consider lend as the X coordinate. And with as the Y coordinates, we can just say that area equals X multiplied by Y. And we know that y equals 4 minus X2. Which means that area equals x multiplied by 4 minus X squad. So we now have the area function in terms of X. And we can distribute. S 4 X minus XQ. What we're going to do is simply optimize this function by finding the derivative of A of X. Which is the derivative of 4 x minus X cubed. And this derivative can be found by. Applying the power rule for differentiation, so we get 4 minus 3 x 2. Now, let's identify the critical points by setting our derivative equal to 0. Which means that 4 minus 3 X 2 is equal to 0. If we rearrange this equation, we get 4 equals 3 x 2. And X2 is equal to. 4 divided by 3. From here we can find X by taking square root of both sides, and let's recall that square root of X is positive or negative X, so we're going to have two solutions. So X is positive or negative. Square root of 4/3. Which is positive or negative. Of 2 divided by a square root of 3. And now we can rationalize the denominator, so X is positive or negative of 2 square root of 3 divided by 3. Now, because we have a rectangle in the first quadrant, this means that X is greater than 0. So the only critical point is 2, square root of 3 divided by 3. Now we want to understand if this is a local minimum or a local maximum. So we have to apply the 2nd derivative test. And we're going to find the 2nd derivative. Which is the derivative of 4 minus 3 x 2. Applying the power rule. We get -6 X for the derivative. The derivative of 4 is 0, right? It's a constant. And now let's identify the 2nd derivative at the critical point. So 2 square roots of 3 divided by 3. This would be equal to -6 multiplied by that, which gives us -12. Square root of 3 divided by 3. Which is negative. 4 square roots of 3. And this is negative. According to the 2nd derivative test, our function is concave down at this critical point, meaning we have a maximum. So we have verified that this critical point corresponds to the maximum. And now let's identify the largest area, which is area at 2 square root of 3. Divided by 3. We can substitute for this x value. Which gives us 2 square root of 3. Divided by 3. Multiplied by Or minus. 2 square roots of 3 divided by 3. Squared Now let's evaluate the result. We're going to get 2 square roots of 3 divided by 3. Multiplied by or minus. 2 square 4, square root of 3 square is 3. So we got 4 Multiplied by 3 divided by 3 squad, which is 9. Finding the common denominator, we get 2 square roots of 3 divided by 3, multiplied by 4 multiplied by 9 is 36. We are subtracting 12. And dividing by 9. So now, let's identify the result. We are going to get 2 square roots of 3 divided by 3. Multiplied by 24. Divided by 9. We can simplify as to square roots of 3 divided by 3, multiplied by, now 24 is divisible by 3, which gives us 8, and 9 is divisible by 3, which gives us 3. So now 2 multiplied by 8. Its a 16. We have square root of 3 in the numerator, and in the denominator 3 multiplied by 3 gives us 9. So the largest possible area of this rectangle is 16 square roots of 3 divided by 9 units. That's it. Thank you for watching.

41. Among all triangles in the first quadrant formed by the x-axis, the y-axis, and tangent lines to the graph of y=3x-x^2, what is the smallest possible area?

Key Concepts

Tangent Line to a Curve

Area of a Triangle

Optimization in Calculus

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