A rectangular region is completely covered by identical square tiles. Each tile has area . The length of the rectangular region is 2.125 times its width. The width is equal to . Which choice is the value of ?

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Super-Hard SAT Math Question 80 | Free SAT Prep | Aniko

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Question 80·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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A rectangular region is completely covered by $6{,}664$ identical square tiles. Each tile has area $k$ .

The length of the rectangular region is 2.125 times its width. The width is equal to $x\sqrt{k}$ .

Which choice is the value of $x$ ?

Translate the word description into an area equation: total area from tiles equals length times width. Use the length-to-width multiplier to write area as a constant times $W^2$ , then substitute $W=x\sqrt{k}$ so the radical disappears when squared. After canceling $k$ , solve for $x^2$ first and take the square root at the end.

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Start with total area

Write the rectangle’s area in terms of the number of tiles and the area of each tile.

Use the length-to-width relationship

If the width is $W$ , express the length in terms of $W$ and write the rectangle’s area as a multiple of $W^2$ .

Substitute the width expression carefully

Replace $W$ with $x\sqrt{k}$ and remember that $(x\sqrt{k})^2=x^2k$ .

Desmos Guide

Graph both sides of the equation

Enter $y=(17/8)x^2$ and enter $y=6664$ as a second equation.

Find the intersection

Click the intersection point of the two graphs to see the $x$ -coordinates where they meet.

Choose the meaningful solution

Use the positive $x$ -value from the intersection (since $x$ represents a length factor) as the value of $x$ .

Step-by-step Explanation

Relate tile area to rectangle area

Because the region is completely covered by $6{,}664$ tiles and each tile has area $k$ , the rectangle’s area is

6{,}664k.

Write the rectangle’s area using width and length

Let the width be $W$ . Then the length is $2.125W$ , so the rectangle’s area is

(2.125W)(W)=2.125W^2.

Substitute $W=x\sqrt{k}$ and cancel $k$

Substitute $W=x\sqrt{k}$ :

6{,}664k = 2.125(x\sqrt{k})^2 = 2.125x^2k.

Since $k>0$ , divide both sides by $k$ :

6{,}664 = 2.125x^2.

Solve for $x$

Rewrite $2.125$ as a fraction: $2.125=\frac{17}{8}$ .

6{,}664 = \frac{17}{8}x^2 \quad\Rightarrow\quad x^2 = 6{,}664\cdot \frac{8}{17}.

Compute:

6{,}664\cdot \frac{8}{17}= \frac{53{,}312}{17}=3{,}136.

So $x=\sqrt{3{,}136}=56$ .

Therefore, the correct choice is 56.

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Step-by-step Explanation

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