Science:Math Exam Resources/Courses/MATH104/December 2013/Question 01 (n)

MATH104 December 2013

  •  Q1 (a)  •  Q1 (b)  •  Q1 (c)  •  Q1 (d)  •  Q1 (e)  •  Q1 (f)  •  Q1 (g)  •  Q1 (h)  •  Q1 (i)  •  Q1 (j)  •  Q1 (k)  •  Q1 (l)  •  Q1 (m)  •  Q1 (n)  •  Q2 (a)  •  Q2 (b)  •  Q2 (c)  •  Q2 (d)  •  Q2 (e)  •  Q2 (f)  •  Q2 (g)  •  Q3  •  Q4  •  Q5  •  Q6 (a)  •  Q6 (b)  •  Q6 (c)  •  

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Question 01 (n)
If the base $b$ of a triangle is increasing at a rate of 3 centimetres per second while its height $h$ is decreasing at a rate of 3 centimetres per second, which of the following must be true about the area $A$ of the triangle? (A) $A$ is always increasing. (B) $A$ is always decreasing. (C) $A$ is decreasing only when $b<h$. (D) $A$ is decreasing only when $b>h$. (E) $A$ 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!

Hint
The area of a triangle is $A={\frac {bh}{2}}.$ Try evaluating ${\frac {dA}{dt}}$ in terms of $b$ and $h$ (both functions of $t$) to get some insight.

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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. The area of a triangle is given by $A={\frac {bh}{2}}.$ 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. $A'={\frac {1}{2}}\left(bh'+hb'\right).$ Since b' = 3 and h' = -3, A' becomes $A'={\frac {3}{2}}\left(h-b\right).$ 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 $0>{\frac {3}{2}}\left(h-b\right)\quad \Rightarrow \quad b>h.$ Thus, the correct answer is (D).

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