Science:Math Exam Resources/Courses/MATH104/December 2013/Question 01 (n)
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Question 01 (n)
If the base of a triangle is increasing at a rate of 3 centimetres per second while its height is decreasing at a rate of 3 centimetres per second, which of the following must be true about the area of the triangle?
(A) is always increasing.
(B) is always decreasing.
(C) is decreasing only when .
(D) is decreasing only when .
(E) remains constant.
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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
The area of a triangle is Try evaluating in terms of and (both functions of ) to get some insight.
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The area of a triangle is given by
To determine the rate of change of area of the triangle, we evaluate A' by differentiating A with respect to t, recognizing that both b and h are functions of time.
Since b' = 3 and h' = -3, A' becomes
By inspecting this equation we can determine the answer. Clearly A is not always increasing nor always decreasing nor constant since the both the sign and the value of A' depend on the values of b and h. So (A), (B), and (E) are not true. If A is decreasing, then A' < 0 and so
Thus, the correct answer is (D).