Question 131·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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A rectangular mosaic has a total area equal to the combined area of $5{,}175$ congruent square tiles. Each tile has area $k$ square units. The length of the mosaic is $1.44$ times its width. If the width of the mosaic is equal to $x\sqrt{k}$ units, which choice is the value of $x$ ?

A tiled rectangle points to area: the number of tiles times each tile’s area equals length times width. Express the longer side using the given multiplier, then cancel the tile-area factor before solving for the number in the width. A positive-only Desmos regression handles the square without admitting a negative length. Remember that changing both side lengths changes the area by the square of that factor.

Hints

Hint 1
Each tile has area $k$ , and there are $5{,}175$ congruent tiles. Multiplying the area of one tile by the number of tiles gives the mosaic’s total area. What expression represents that total?
Hint 2
A rectangle’s area is length times width. The length is $1.44$ times the width, so multiply $x\sqrt{k}$ by $1.44x\sqrt{k}$ and set that product equal to the tile area.
Hint 3
A square root multiplied by itself gives the number under the root. After simplifying $\sqrt{k}\cdot\sqrt{k}$ , you can cancel $k$ because a tile’s area is positive. Which sign of $x$ can represent a width?

Step-by-step

Equate the two areas and solve
Step 1
Find the area from the tiles
Each of the $5{,}175$ congruent tiles has area $k$ , so the mosaic’s area is $A=5{,}175k$ .
Step 2
Write the area using the side lengths
The length is $1.44$ times the width $x\sqrt{k}$ , so it is $1.44x\sqrt{k}$ . A rectangle’s area is length times width. Set that product equal to the tile area: $(1.44x\sqrt{k})(x\sqrt{k})=5{,}175k$ .
Step 3
Simplify the product
A square root satisfies $\sqrt{k}\cdot\sqrt{k}=k$ . Multiply the factors on the left, keeping both copies of $x$ : $1.44x^2k=5{,}175k$ .
Step 4
Cancel the tile-area factor
A tile has positive area, so $k>0$ . Divide both sides by $k$ : $1.44x^2=5{,}175$ . The tile area cancels, leaving an equation for the number multiplying $\sqrt{k}$ .
Step 5
Solve and verify the exact choice
Type $1.44x_1^2\sim5175\{x_1>0\}$ in Desmos. The $\sim$ asks Desmos to solve for $x_1$ , and the restriction keeps the positive value required for a width. Under PARAMETERS, it shows $x_1\approx59.948$ . To check the matching radical choice exactly, type $1.44\left(\frac{25\sqrt{23}}{2}\right)^2$ on the next line; Desmos gives $5{,}175$ . So $x=\frac{25\sqrt{23}}{2}$ , the coefficient multiplying $\sqrt{k}$ in the width. Choice B.

Question 131·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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