MHB How Do You Find the Length x in a Geometric Problem with Shaded Areas?

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To find the length x in the geometric problem with a shaded area of 1200 cm², the areas of a triangle and rectangle are considered. The triangle's area is initially set up as A_triangle = (1/2)(x-1)(x), but it is corrected to A_triangle = (1/2)x². The equation combining both areas is A_rectangle + A_triangle = 2400, leading to the equation (1/2)x² + x = 1200. Solving this quadratic equation yields x = 48 cm as the valid solution. The final answer is x = 48 cm.
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Find the length x if the shaded area is 1200 cm^2

I tried to solve this is what I get

since $A_{triangle}=(\frac{1}{2})({x-1})(x)$

and $A_{rectangle}=x$

$A_{rectangle}+A_{triangle}=2400$

Is the set-up of my equation correct?
 

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Why are you using $x-1$ in the area of the triangle?
 
MarkFL said:
Why are you using $x-1$ in the area of the triangle?

I see it. $A_{TRI}=\frac{1}{2}(x^2)$

now I will have

$\frac{1}{2}(x^2)+x=2400$

solving for x $(x-48)(x+50)=0$

$x=48 in.$ :D
 
Well, you actually have:

$$\frac{1}{2}x^2+x=1200$$

or:

$$x^2+2x-2400=0$$

$$(x+50)(x-48)=0$$

Discarding the negative root, we then find:

$$x=48\text{ cm}$$
 

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