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 $x$ ?

A rectangle completely covered by identical tiles is an area problem: multiply the number of tiles by one tile’s area. Then write the rectangle’s area as length times width and set the two expressions equal. If the length is a multiple of the width, that multiplier belongs in the dimension product. Solve the resulting equation in Desmos, but don’t mistake the width squared for the requested value.

Hints

Hint 1
The area is the space covered. Since the tiles cover the rectangle completely, multiply the number of tiles by one tile’s area. What expression gives the area of the whole rectangle?
Hint 2
The length is a multiple of the entire width, not of one tile’s area. Multiply the given expression for width by the length multiplier, then use length times width for the rectangle’s area.
Hint 3
Multiplying the two dimensions gives a factor of $\sqrt{k}\cdot\sqrt{k}=k$ . Since a tile has positive area, you can cancel $k$ from both sides. The remaining equation involves $x^2$ , so what must you do to find $x$ ?

Step-by-step

Equate the tile area and rectangle area
Step 1
Find the area from the tiles
The area is the space covered. All $6{,}664$ tiles have area $k$ , and they cover the rectangle completely, so multiply the tile count by $k$ :
$A=6{,}664k.$
Step 2
Write the length in terms of the width
Write $L$ for the rectangle’s length. It is $2.125$ times the entire width, $x\sqrt{k}$ , so:
$L=2.125(x\sqrt{k}).$
Step 3
Equate the two expressions for area
A rectangle’s area is length times width. Counting the tiles and multiplying the dimensions must give the same area. Substitute the dimensions and set their product equal to the tile total:
$6{,}664k=2.125(x\sqrt{k})(x\sqrt{k}).$
Step 4
Multiply the matching factors
The two $x$ factors make $x^2$ , and $\sqrt{k}\cdot\sqrt{k}=k$ . Rewrite the dimension product:
$6{,}664k=2.125x^2k.$
Step 5
Cancel the tile area
A tile has positive area, so $k>0$ . Divide both sides by $k$ :
$6{,}664=2.125x^2.$
Step 6
Solve for the positive width multiplier
The width $x\sqrt{k}$ must be positive, so keep $x>0$ . In Desmos, type $2.125x_1^2\sim6664\{x_1>0\}$ . The $\sim$ tells Desmos to solve for $x_1$ rather than graph an equation in $x$ . Under PARAMETERS, it shows $x_1=56$ . So the width is $56\sqrt{k}$ , and $x=56$ . Choice D.

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

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