27. water (w) was added to the graduated cylinder containing liquid c. objects a and d were then dropped into the cylinder. which diagram most accurately shows the resulting arrangement of these substances? a) diagram of graduated cylinder with c (top), w (bottom), a and d at bottom, b) diagram of graduated cylinder with c (top), w (bottom), a floating at w-c interface, d at bottom, c) diagram of graduated cylinder with w (top), c (bottom), a floating at top of w, d in c, d) diagram of graduated cylinder with w (top), c (bottom), d floating at w-c interface, a at bottom of c

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# Brief Explanations: To solve this, we analyze the density relationships: 1. Liquid C and water (W) are immiscible (since they form separate layers), so we first determine which is denser. In the diagrams, C is below W in some, meaning $ ho_C > ho_W $ (density of C is greater than water). 2. For objects A and D: - Object D: In diagram B, D is at the bottom of W (but W is above C, which is denser than W). Wait, no—wait, the correct logic is: If $ ho_D > ho_C $, D would sink through W and C? No, wait, the layers: C and W are separate. So first, C and W: since in the initial setup, when water is added to C, if C is denser than W, C will be at the bottom, W on top (like oil and water, but reversed if C is denser). Wait, in the diagrams, in option B: C is top layer, W is middle, D at bottom of W, A floating in W? No, let's re-express: Wait, the key is: - Liquid C and Water (W) are immiscible, so they form two layers. The denser liquid will be at the bottom. So if C is denser than W, C is below W? No, wait, in the diagrams, in option B: C is the top layer, W is middle, D at the bottom of W (so D is denser than W but less than C? No, wait, let's look at the objects: Object A: In option B, A is floating in W (so $ ho_A < ho_W $), Object D: in option B, D is at the bottom of W (so $ ho_D > ho_W $), but since C is above W, that would mean $ ho_C < ho_W $, but then C would float on W. Wait, no—maybe I got the layers reversed. Wait, when you add water to liquid C, if C is denser than water, C will be at the bottom, water on top. So C (denser) is below W (less dense). Then: - Object A: If A floats in W, $ ho_A < ho_W $. - Object D: If D is denser than C, it would sink through W into C. If D is denser than W but less than C, it would sink to the bottom of W (but above C). Wait, no—let's check the diagrams: Option B: - Top layer: C (density $ ho_C $) - Middle layer: W ( $ ho_W $) - Object A: floating in W (so $ ho_A < ho_W $) - Object D: at the bottom of W (so $ ho_D > ho_W $) but since C is above W, that would mean $ ho_C < ho_W $, so C floats on W. Then D is denser than W but less than C? No, that can't be. Wait, maybe the correct layer is C below W (C denser than W). Then: Option B: Wait, no, let's look at the diagrams again. The correct logic is: 1. Immiscible liquids: C and W form two layers. Denser liquid is at the bottom. So if C is denser than W, C is below W. 2. Object A: If A floats in W, $ ho_A < ho_W $. 3. Object D: If D is denser than C, it sinks to the bottom (through W and C). If D is denser than W but less than C, it sinks to the bottom of W (above C). Now, looking at the options: - Option A: A and D at the bottom (below C and W), so $ ho_A, ho_D > ho_C $ (since C is above W, which is above A and D? No, A and D are at the bottom, C is middle, W is top? No, the diagram A has C (top), W (middle), A and D at bottom (so below W and C). So $ ho_A, ho_D > ho_C $, but then C would be less dense than W? No, this is confusing. Wait, the correct answer is B. Let's re-express: - Liquid C and W are immiscible. Let's assume C is denser than W (so C is below W? No, in diagram B, C is above W. Wait, maybe C is less dense than W, so C floats on W (like oil on water). Then: - Object A: Floats in W (so $ ho_A < ho_W $) → so A is in W, floating. - Object D: Denser than W but less than C? No, D is at the bottom of W (so $ ho_D > ho_W $) but since C is above W, $ ho_C < ho_W $, so D is denser than W but less than C? No, that would mean D is between W and C, but in diagram B, D is at the bottom of the cylinder (below W). Wait, no—diagram B: C (top), W (middle), A (floating in W), D (at the bottom of the cylinder, below W). So D is denser than W and denser than C (since it's below C? No, C is above W, so D is below W, which is below C? No, the cylinder's layers: C is top, W is middle, D is at the bottom (so below W), A is in W. So D is denser than W, and since it's at the bottom, denser than C? No, C is above W, so C is less dense than W. Then D is denser than W, so denser than C. But A is floating in W, so less dense than W. This matches: - C (less dense than W) → top layer. - W (middle layer). - A: $ ho_A < ho_W $ → floats in W. - D: $ ho_D > ho_W $ (and since it's at the bottom, $ ho_D > ho_C $ as well, but since C is less dense than W, D just needs to be denser than W to sink through W to the bottom (since C is on top, but D is below W, so C is above W, D is below W—so the order is C (top), W (middle), D (bottom), A (in W). This makes sense because: - Immiscible liquids: C (less dense) on top of W (more dense). - A: less dense than W → floats in W. - D: more dense than W → sinks to the bottom (below W, since W is above D? Wait, no—if W is in the middle, C on top, W in middle, then D at the bottom (below W) means D is denser than W, so it sinks through W to the bottom. A is less dense than W, so it floats in W. This matches diagram B. Other options: - Option A: A and D at the bottom (below C and W), so they are denser than C and W. But A is at the bottom, so $ ho_A > ho_C $ and $ ho_A > ho_W $, but then A would sink through W and C, but in diagram A, A is at the bottom, C is above W, W above A—so C is less dense than W, W less dense than A. But then A would sink through W and C, which is possible, but A is shown with D at the bottom. But A is supposed to float? No, in the problem, we don't know if A floats, but the diagram B shows A floating in W, which is more logical. - Option C: D is in C (bottom layer), A is floating in W (top layer). But C is below W, so C is denser than W. Then D is in C, so $ ho_D > ho_C $, and A is in W, $ ho_A < ho_W $. But then W is above C, so W is less dense than C. But in diagram C, W is above C, so that's correct (C denser than W). But A is floating in W (so $ ho_A < ho_W $), D is in C ( $ ho_D > ho_C $ ). But the problem says objects A and D were dropped into the cylinder with C and W. If C is denser than W, then W would be on top, C below. Then dropping A (less dense than W) would float in W, dropping D (denser than C) would sink to the bottom (through W and C). But diagram C shows D in C (bottom layer) and A in W (top layer), but the position of W and C: in diagram C, W is above C, so that's correct (C denser). But the problem is the initial setup: water (W) was added to the graduated cylinder containing liquid C. So initially, C is in the cylinder, then W is added. So if C is denser than W, W will be on top of C (like adding water to mercury—water is on top). So W is top, C is bottom. Then: - A: dropped into W—if $ ho_A < ho_W $, floats in W. - D: dropped into W—if $ ho_D > ho_C $, sinks through W into C (bottom layer). But diagram C shows W above C, D in C (bottom), A in W (top). But in diagram C, the W is above C, but the D is in C (correct, if D is denser than C), and A is in W (correct, if A is less dense than W). But wait, in diagram B, W is in the middle, C on top. So which is correct: is C denser than W or less? Wait, the key is that when you add water to liquid C, if they are immiscible, the denser one is at the bottom. So if after adding W to C, the layers are C (top) and W (bottom), that would mean $ ho_C < ho_W $ (C is less dense, so floats on W). Wait, no—water is denser than oil, so oil (less dense) floats on water (more dense). So if C is like oil (less dense than W), then C is on top, W is below. Then: - A: dropped into W—if $ ho_A < ho_W $, floats in W (middle layer). - D: dropped into W—if $ ho_D > ho_W $, sinks to the bottom (below W), since W is below C. Which matches diagram B: C (top), W (middle), A (floating in W), D (bottom of cylinder, below W). Yes, this makes sense. So the correct diagram is B. # Answer: B

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