Science:Math Exam Resources/Courses/MATH110/April 2017/Question 09

MATH110 April 2017

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[hide] Question 09

A gardener has a patch of land in the shape of a right-angled triangle. The two sides of the garden next to the right-angle have lengths $3$ metres and $4$ metres, respectively. The gardener wants to build a rectangular flower bed on the land so that one corner of the bed is on the right-angle, and the opposite corner of the rectangle lies on the hypotenuse. Find the dimensions of the largest such flower bed that can be built inside the garden.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

[show] Hint 1
What general type of problem is this? You're trying to maximize the area of the rectangle, so this is an optimization problem. You should try to draw a picture in order to get a better handle on what the question is asking. Your picture should look something like the following.

[show] Hint 2
The width of the rectangle in the picture is x. What is the height as a function of x? Try using similar triangles in order to figure this out.

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

[show] Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We continue from the hint. Let y be the height of the rectangle in the picture. We would like to solve for y in terms of x. The small triangle on the bottom right is similar to the big triangle with dimensions 3 meters and 4 meters. The height of this triangle is y, and the width of this triangle is 3-x. We therefore have the following similarity equation: ${\frac {3}{3-x}}={\frac {4}{y}}.$ We multiply both sides by y and by 3-x: $3y=4(3-x)$ And now, we divide by 3: $y={\frac {4(3-x)}{3}}$ Because the rectangle has width x and height y, we have that the area of the rectangle is $x{\frac {4(3-x)}{3}}$ We multiply the x into the numerator: ${\frac {4x(3-x)}{3}}$ and multiply the 4x into the (3-x) term: ${\frac {12x-4x^{2}}{3}}$ and rewrite this as the difference of two fractions: $4x-{\frac {4}{3}}x^{2}.$ So we have that the area, A, is $A=4x-{\frac {4}{3}}x^{2}.$ Because we want to maximize the area, we should take the derivative of A and set it equal to zero: ${\begin{aligned}{\frac {dA}{dx}}&=4-{\frac {8}{3}}x\\0&=4-{\frac {8}{3}}x\\{\frac {8}{3}}x&=4\\x&={\frac {12}{8}}\\x&={\frac {3}{2}}\end{aligned}}$ It remains to compute the y-value for each x. We computed that $y={\frac {4(3-x)}{3}}$ If we plug in x = 3/2, we get ${\begin{aligned}y&={\frac {4(3-3/2)}{3}}\\y&={\frac {4(3/2)}{3}}\\y&={\frac {6}{3}}\\y&=2.\end{aligned}}$ This gives the dimensions $x=3/2;y=2$ We should also check the endpoints of our domain. The smallest x can possibly be is zero, in which case the area is obviously zero. Similarly, the largest x can be is 3, in which case the area is also zero. Therefore, the point $x=3/2;y=2$ is a genuine maximum. Answer: $\color {blue}x=3/2;y=2$