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The math problem

I am not sure how to answer this, it is for a quiz that I am doing and my teacher hasn't specified how to answer this type of problem. I've tried distributing the $x$ out and I got $(x^2-4x+10)/2$ as my minimum area but it's saying I am wrong. I even tried putting an inequality of $>4$ but that's also wrong. I have no idea what to do with this math problem.

Manaswini Datta
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    Well, if we let the function $A(x)$ be the area of the triangle - given as $1/2 \cdot b \cdot h$ - then

    $$A(x) = 5 - \frac{1}{2}x^2 - 2x$$

    In calculus, we learn that we can use the first derivative test to find minima, i.e. wherever $A'(x) = 0$ there is a minimum or maximum, and trying $x$ in $A(x)$ on just either side of the $x$ that gives $A'(x) = 0$ can help us see if it's a minimum or maximum. (If it's a minimum, for $x$ to the nearby left and right, $A(x)$ will be greater than what you get at the the $x$ that gives a zero derivative.)

     – PrincessEev Nov 02 '18 at 08:52
  • In other words, you are not seeking the formula for the area of the triangle. You're seeking, first, to find which $x$ gives the smallest possible area. Then you plug that in to $A(x)$ which will give you that minimum area - and thus, your answer. – PrincessEev Nov 02 '18 at 08:54
  • Welcome to MSE. Please choose your tags with care. Your question is clearly not about education. – José Carlos Santos Nov 02 '18 at 08:54
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    Express the area as a function of $x$, then differentiate that function so as to find its stationary points. Upon finding the stationary points, determine whether they are maxima or minima, and for the minimum, take the corresponding value of $x$, plug that back into your equation for the area, and hey presto! – Sam Streeter Nov 02 '18 at 08:54
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    You should "complete the square". the area satisfies $\frac12(x^2 - 4x + 10) = \frac12((x-2)^2 + 6) \ge \frac12(0^2 + 6) = 3$. So the minimum is $3$ by putting $x = 2$. – achille hui Nov 02 '18 at 09:04
  • @achillehui that is almost exactly what went through my mind - no need to complicate things by multiplying everything by $\frac 12$, use a simple completion of the square rather than using a formula. Less chance to go wrong. More insight into what is going on. – Mark Bennet Nov 02 '18 at 09:49

1 Answers1

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Use $A = \frac{1}{2}bh$ to find the function of the area.

$$A = \frac{1}{2}(x)\bigg(\frac{10}{x}+x-4\bigg)$$

$$A = \frac{1}{2}x^2-2x+5$$

Think of this as a function - a quadratic function. The maxima/minima of a function can be found where its first derivative is equal to $0$. This is a quadratic function where the coefficient of the leading term ($a$) is positive. Hence, we can confirm it must be the minimum.

$$A’ = x-2$$

Now, set $A’ = 0$ and see what you get for $x$. Evaluate $A$ (the original function) for that value of $x$ you get.

Edit: If you aren’t familiar with calculus, you could just go with finding the vertex of function $A$. The vertex of the function is at the point $(h, k)$.

$$A = \frac{1}{2}x^2-2x+5$$

$$h = -\frac{b}{2a}$$

$h$ is the $x$-coordinate of the vertex of function $A$. Evaluate it, and just plug in that value in the original function to find $k$, or the $y$-coordinate of the vertex of function $A$. That value of $k$ will be the minimum output, or in this case, the minimum area.

Yet again, you can immediately find $k$ without calculating $h$ first by using a general formula.

$$k = c-\frac{b^2}{4a}$$

KM101
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