Uche is a cartographer. He picks a scale to fit a map of India onto a page of an atlas. The page is 12 by 12 inches, with 0.75 inch margins on all 4 sides. India measures 3,214 kilometers from north to south and 2,933 kilometers from west to east. Uche wants the longest dimension of India to fit exactly in between the margins of the page. If k is the number of kilometers per inch in Uche’s scale, which equation best models the situation?
The correct answer and explanation is:
To model the situation, first calculate the effective area of the page where the map can fit. Since there are 0.75-inch margins on all four sides, the usable area for the map is reduced. The page measures 12 by 12 inches, but the margins subtract from both sides.
The length of the usable page is:
12 inches – 2 × 0.75 inches = 10.5 inches
This means that the longest dimension of India, which is its north-to-south distance of 3,214 kilometers, must fit into the 10.5-inch space on the map.
Let the scale of the map be kk, which represents the number of kilometers per inch. This scale means that for each inch on the map, it represents kk kilometers in real life.
Now, we set up the equation. The longest dimension of India is 3,214 kilometers, and this must be scaled down to fit the 10.5-inch length on the map. Therefore, the relationship between the real-world distance and the scaled distance is: 3214k=10.5\frac{3214}{k} = 10.5
Here, 3214k\frac{3214}{k} represents the number of inches on the map that correspond to the 3,214 kilometers from north to south. Solving this equation for kk: k=321410.5k = \frac{3214}{10.5}
Thus, kk is the number of kilometers per inch in Uche’s scale. This equation models the situation where the longest dimension of India fits exactly between the margins on the page.
In summary, the equation that best models the situation is 3214k=10.5\frac{3214}{k} = 10.5. By solving this equation, Uche can find the scale factor kk, which will allow him to fit the map of India onto the page of the atlas.