Question

a paper cup is 7 in. tall with 3 in. of that height being the rim. the cups nest together with the non - rim part (4 in.) fitting in a cup below. write an expression for the total height of n nested cups using 4 to represent each of the unseen nested portions of the cups.
(simplify your answer.)

Explanation:

Step1: Analyze single cup height

A single cup is 7 in. tall, with rim 3 in. and non - rim (unseen when nested) 4 in. But when nesting, the first cup has full height 7, and each additional cup adds (7 - 4) = 3 in. (since 4 in. is hidden in the below cup). Wait, no, re - reading: each unseen nested portion is 4. Wait, the problem says "using 4 to represent each of the unseen nested portions of the cups". So for n cups: the first cup contributes 7 in. (total height of first cup). For each of the remaining (n - 1) cups, the visible part? Wait, no. Wait, when you nest cups, the total height is the height of the first cup plus (n - 1) times the height that each additional cup adds. Since each unseen portion is 4, the height added by each additional cup is (7 - 4) = 3? Wait, no, maybe another approach. Let's think: the first cup has height 7. Each cup after the first, when nested, has 4 in. hidden, so the height contributed by each of the (n - 1) cups is (7 - 4) = 3? Wait, no, the problem says "write an expression for the total height of n nested cups using 4 to represent each of the unseen nested portions of the cups". So total height = height of first cup + (n - 1) * (height of a cup - unseen portion). Wait, no, the unseen portion is 4. So the height added by each cup after the first is (7 - 4) = 3? Wait, no, let's re - express. The total height H of n nested cups: the first cup is 7 in. For each of the other (n - 1) cups, when nested, the part that contributes to the total height is (7 - 4) = 3 in. (because 4 in. is hidden in the cup below). So H = 7+3(n - 1). Simplify this: 7 + 3n-3=3n + 4. Alternatively, another way: each cup has a height of 7, but when nested, (n - 1) cups have 4 in. hidden. So total height = 7n-4(n - 1). Let's check: 7n-4n + 4=3n + 4. Yes, that works. Let's verify with n = 1: 3(1)+4 = 7, correct. n = 2: 3(2)+4 = 10. First cup 7, second cup adds 3 (7 - 4), 7+3 = 10, correct. So the expression is 3n + 4 or 7+3(n - 1) or 7n-4(n - 1).

Step2: Derive the expression

We can think of it as: the first cup contributes 7 inches. For each of the remaining (n - 1) cups, since 4 inches are unseen (fitting into the below cup), the visible (added) height per cup is 7 - 4 = 3 inches. So total height H=7 + 3(n - 1). Simplifying:

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Or, another approach: each of the n cups has a height of 7, but (n - 1) cups have 4 inches hidden (because only the top cup has no hidden part? Wait, no, the bottom cup has no hidden part? Wait, no, when you nest n cups, the number of hidden portions is (n - 1), because each cup except the bottom one has 4 inches hidden in the cup below. So total height = sum of heights of all cups - sum of hidden portions. Sum of heights of all cups is 7n. Sum of hidden portions is 4(n - 1). So H = 7n-4(n - 1)=7n-4n + 4 = 3n + 4.

Answer:

The expression for the total height of \( n \) nested cups is \( 3n + 4 \) (or equivalent forms like \( 7 + 3(n - 1) \) or \( 7n-4(n - 1) \)).