Question

furniture marcus is assembling a leaning bookshelf with the dimensions shown in the diagram.

diagram of a leaning bookshelf with vertical segments of 12 in. each, a slant side of 50 in., and a segment labeled x in.

if the shelves are parallel and spaced 12 inches apart, what is the value of x?

x = blank box

Explanation:

Step1: Determine total height

The vertical side has 4 segments of 12 in, so total height \( h = 4\times12 = 48 \) in. Wait, no, the slant side is 50 in. Wait, the shelves are parallel, so by the Basic Proportionality Theorem (Thales' theorem), the segments on the slant side are proportional to the vertical segments. Wait, the vertical segments: from bottom to top, the first shelf (from bottom) is at 12 in, second at 24, third at 36, fourth at 48? Wait, no, the diagram: the vertical side has four 12-in segments, so total vertical length is \( 12\times4 = 48 \) in? Wait, no, the slant side is 50 in. Wait, the x is the segment from the third shelf (from bottom) to the bottom? Wait, no, let's see: the vertical side has four 12-in parts, so the distance from the bottom to the top vertical is \( 12\times4 = 48 \) in. The slant side is 50 in. The shelves are parallel, so the triangles formed are similar. So the ratio of x to 50 should be equal to the ratio of (12×3) to 48? Wait, no. Wait, the vertical segments: from bottom to the first shelf (bottom shelf) is 12 in, to the second is 24, to the third is 36, to the top is 48. Wait, the slant side: the total slant is 50. The segment x is from the third shelf (counting from bottom: bottom is 1st, then 2nd, 3rd, top 4th) to the bottom? Wait, no, the diagram: the vertical has four 12-in, so the distance from bottom to top vertical is \( 12\times4 = 48 \) in. The slant side is 50 in. The shelves are parallel, so the triangles are similar. So the ratio of the length of the slant segment (x) to the total slant (50) is equal to the ratio of the vertical segment (from bottom to the shelf) to the total vertical (48). Wait, no, maybe the vertical from bottom to the shelf where x starts is \( 12\times3 = 36 \) in? Wait, no, let's count the number of 12-in segments below the x segment. Wait, the x is between the third shelf (from bottom) and the bottom? Wait, the diagram: the vertical has four 12-in: bottom to first shelf (12), first to second (12), second to third (12), third to top (12). So the distance from bottom to third shelf is \( 12\times3 = 36 \) in, and total vertical is \( 12\times4 = 48 \) in. So by similar triangles, \( \frac{x}{50} = \frac{36}{48} \). Simplify \( \frac{36}{48} = \frac{3}{4} \). Then \( x = 50\times\frac{3}{4} = 37.5 \)? Wait, no, that can't be. Wait, maybe I got the ratio reversed. Wait, the total vertical is 48, and the slant is 50. Wait, maybe the vertical from bottom to the top is 48, and the slant is 50. The shelves are parallel, so the segments on the slant are proportional to the vertical segments. Wait, the x is the length from the third shelf (from bottom) to the bottom. Wait, the number of 12-in segments below x: 3? Wait, no, let's look at the diagram again. The vertical has four 12-in: bottom to first shelf (12), first to second (12), second to third (12), third to top (12). So the distance from bottom to third shelf is \( 12\times3 = 36 \) in, and from bottom to top is \( 12\times4 = 48 \) in. So the ratio of 36 to 48 is 3/4. So the slant segment x should be 50*(3/4) = 37.5? Wait, but 48 and 50: 48 is vertical, 50 is slant. Wait, maybe the vertical is 48, slant is 50, so the ratio of vertical to slant is 48/50 = 24/25. Then the vertical segment (36) corresponds to slant segment x: 36 = (24/25)x? No, that's not. Wait, similar triangles: the smaller triangle (from bottom to third shelf) and the larger triangle (from bottom to top) are similar. So the ratio of corresponding sides is equal. So (length of smaller vertical)/(length of larger vertical) = (length of smaller slant)/(length of larger slant). So 36/48 = x/50. Simplify 36/48 = 3/4. So x = 50*(3/4) = 37.5. Wait, but 48 and 50: 48 is vertical, 50 is slant. Wait, maybe the vertical is 48, slant is 50, so the ratio of slant to vertical is 50/48 = 25/24. Then the slant segment x is (25/24)*36 = 25*1.5 = 37.5. Yes, that makes sense. So x = 37.5. Wait, but let's check again. The vertical has four 12-in, so total vertical height is 48 in. The slant side is 50 in. The shelves are parallel, so the triangles formed by the vertical, slant, and shelves are similar. So the ratio of the slant segment (x) to the total slant (50) is equal to the ratio of the vertical segment (from bottom to the shelf) to the total vertical (48). The vertical segment from bottom to the shelf (where x starts) is 12*3 = 36 in (since there are three 12-in segments below that shelf: bottom to first, first to second, second to third). So 36/48 = x/50. Cross-multiplying: 48x = 36*50. 48x = 1800. x = 1800/48 = 37.5. Yes, that's correct.

Step2: Solve for x

Using the proportion from similar triangles: \( \frac{36}{48} = \frac{x}{50} \). Simplify \( \frac{36}{48} = \frac{3}{4} \). Then \( x = 50\times\frac{3}{4} = 37.5 \). Alternatively, cross - multiply: \( 48x = 36\times50 \), \( 48x = 1800 \), \( x=\frac{1800}{48}=37.5 \).

Answer:

\( 37.5 \)