Question

elvis has a triangular shelf system in his shower. the total height of the shelving system is 98 centimeters, and there are three parallel shelves as shown below. what is a possible height, in centimeters, for the soap box, that can stand upright on the middle shelf? (a) 20 (b) 32 (c) 42 (d) 50

Explanation:

Step1: Set up the equation for total height

The total height is the sum of \(4x\), \(6x\), and \(4x\). So we have the equation \(4x + 6x + 4x=98\).

Step2: Solve for \(x\)

Combine like terms: \(14x = 98\). Then divide both sides by 14: \(x=\frac{98}{14}=7\).

Step3: Find the height of the middle shelf region

The middle shelf's height - related segment is \(6x\), but wait, actually, the soap box stands on the middle shelf, so we need to see the height available. Wait, first, calculate the lengths: \(4x = 4\times7 = 28\), \(6x=6\times7 = 42\), \(4x = 28\). The total height is \(28 + 42+28 = 98\), which checks out. Now, the middle shelf is between the top \(4x\) and bottom \(4x\)? Wait, no, the diagram: the top part is \(4x\), then a middle part with \(6x\) (where the soap box is, \(s\) is the height of the soap box, so \(s\) should be less than \(6x\)? Wait, no, wait, the total height is \(4x + 6x+4x = 14x = 98\), so \(x = 7\). Then \(6x = 42\), but the soap box is on the middle shelf, so its height \(s\) must be less than \(6x\)? Wait, no, maybe I misread. Wait, the three shelves: top shelf, middle shelf, bottom shelf. The distances between them: top to middle is \(6x\)? Wait, no, the sides: the left side has segments \(4x\), \(6x\), \(4x\). So the total height is the sum of these three segments. So the height from top shelf to middle shelf is \(6x\)? Wait, no, maybe the vertical height: the total height is 98, composed of \(4x\) (top to first shelf), \(6x\) (first to second shelf), and \(4x\) (second to third shelf). Wait, no, the diagram: the triangle has three parallel shelves, so the segments on the side are \(4x\), \(6x\), \(4x\), so total side length (height) is \(4x + 6x + 4x = 14x = 98\), so \(x = 7\). Then \(4x = 28\), \(6x = 42\), \(4x = 28\). So the middle shelf is between the \(4x\) (top) and \(4x\) (bottom)? Wait, no, the middle shelf's vertical space: the height available for the soap box (standing upright on middle shelf) would be the height of the middle segment, which is \(6x = 42\)? But wait, the options: 20, 32, 42, 50. Wait, but if the middle segment is \(6x = 42\), but the soap box has to stand upright, so its height must be less than \(6x\)? Wait, no, maybe I messed up the segments. Wait, maybe the top triangle has side \(4x\), then the middle trapezoid has side \(6x\), then the bottom trapezoid has side \(4x\). Wait, no, the total height is the sum of \(4x + 6x + 4x = 14x = 98\), so \(x = 7\). Then \(4x = 28\), \(6x = 42\), \(4x = 28\). So the height from the top shelf to the middle shelf is \(6x = 42\)? No, wait, the top shelf is at height \(4x\) from the top, then the middle shelf is at \(4x + 6x = 10x\) from the top? No, this is confusing. Wait, the total height is 98, so \(4x + 6x + 4x = 14x = 98\), so \(x = 7\). Then the lengths are \(4x = 28\), \(6x = 42\), \(4x = 28\). So the middle shelf is between the \(28\) (top) and \(28\) (bottom)? No, the middle segment is \(6x = 42\), which is the height of the middle region. So the soap box's height \(s\) must be less than \(6x = 42\)? Wait, but option C is 42, but if it's equal, can it stand? Maybe the middle shelf's height available is less than \(6x\)? Wait, no, let's check the options. The possible height must be between \(4x = 28\) and \(6x + 4x = 70\)? No, wait, the top shelf is at \(4x = 28\) from the top, the middle shelf is at \(4x + 6x = 10x = 70\) from the top? No, total height is 98, so from bottom: bottom shelf is at \(4x = 28\) from bottom, middle shelf is at \(4x + 6x = 70\) from bottom? No, this is wrong. Wait, let's re - express: the three segments on the side (height) are \(4x\) (top to first shelf), \(6x\) (first to second shelf), \(4x\) (second to third shelf). So total height: \(4x+6x + 4x=14x = 98\), so \(x = 7\). Then:
- Top to first shelf: \(4x = 28\) cm.
- First to second shelf: \(6x = 42\) cm.
- Second to third shelf: \(4x = 28\) cm.

The soap box is on the middle shelf (second shelf), so it has to stand upright, meaning its height \(s\) must be less than the height of the region it's in, which is the height from first shelf to second shelf, which is \(6x = 42\)? No, wait, the height from the middle shelf to the top shelf is \(6x\)? No, the top shelf is above the middle shelf, with a vertical distance of \(6x\)? Wait, no, the top segment is \(4x\), then the middle segment (between top and middle shelf) is \(6x\), then the bottom segment (between middle and bottom shelf) is \(4x\). So the height available for the soap box (on middle shelf) is the height from middle shelf to top shelf? No, the soap box stands upright, so its height must be less than the height of the space it's in. Wait, the middle shelf is between the top \(4x\) and bottom \(4x\)? No, the lengths are \(4x\), \(6x\), \(4x\), so the order is top: \(4x\), middle: \(6x\), bottom: \(4x\). So the total height is \(4x+6x + 4x = 14x = 98\), \(x = 7\). So \(4x = 28\), \(6x = 42\), \(4x = 28\). So the height of the middle region (where the soap box is) is \(6x = 42\), but the soap box has to stand upright, so its height must be less than \(42\)? Wait, no, maybe the soap box is in the middle shelf, so the height from the middle shelf to the top shelf is \(6x\), and to the bottom shelf is \(4x\). Wait, no, the bottom shelf is at the bottom, so the height from middle shelf to bottom shelf is \(4x = 28\), and to top shelf is \(6x + 4x=10x = 70\)? No, I'm getting confused. Let's think differently. The total height is 98. The three segments are \(4x\), \(6x\), \(4x\). So \(14x = 98\), \(x = 7\). So the lengths are 28, 42, 28. So the middle shelf is between the 28 (top) and 28 (bottom)? No, the middle segment is 42, which is longer than the top and bottom. So the height available for the soap box (on middle shelf) must be less than 42? Wait, but option B is 32, which is less than 42, and option A is 20, also less. Wait, but maybe the soap box's height is between \(4x = 28\) and \(6x = 42\)? Wait, \(4x = 28\), \(6x = 42\). So the height of the soap box must be greater than 28 and less than 42? Let's check the options:
- Option A: 20 < 28, no.
- Option B: 28 < 32 < 42, yes.
- Option C: 42, equal to 6x, maybe not (needs to stand upright, so maybe less than 42).
- Option D: 50 > 42, no.

So the possible height is 32.

Answer:

B. 32