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There are four boys in Paul's family. By the age of nineteen, all of them are over six feet tall. The data table below shows their height at different ages:

\begin{tabular}{|l|l|l|l|l|l|}
\hline & Age 8 & Age 12 & Age 16 & Age 20 & Age 24 \\
\hline Paul & [tex]$4 ft 8 in$[/tex] & [tex]$5 ft 0 in$[/tex] & [tex]$5 ft 11 in$[/tex] & [tex]$6 ft 1 in$[/tex] & [tex]$6 ft 1 in$[/tex] \\
\hline Bruce & [tex]$4 ft 6 in$[/tex] & [tex]$4 ft 11 in$[/tex] & [tex]$5 ft 10 in$[/tex] & [tex]$6 ft 1 in$[/tex] & [tex]$6 ft 1 in$[/tex] \\
\hline Tom & [tex]$4 ft 9 in$[/tex] & [tex]$5 ft 0 in$[/tex] & [tex]$6 ft 0 in$[/tex] & [tex]$6 ft 2 in$[/tex] & [tex]$6 ft 2 in$[/tex] \\
\hline Steve & [tex]$4 ft 7 in$[/tex] & [tex]$4 ft 11 in$[/tex] & [tex]$5 ft 11 in$[/tex] & [tex]$6 ft 1 in$[/tex] & [tex]$6 ft 1 in$[/tex] \\
\hline
\end{tabular}

What can be said about the growth trend for boys in Paul's family?

A. All four boys grew at a steady, consistent rate between the ages of 8 and 24.

B. All four boys had a growth spurt around the age of 20.

C. Between the ages of 12 and 16, all four boys had the biggest jump in height.

D. Between the ages of 8 and 12, all four boys had the biggest jump in height.



Answer :

GinnyAnswer
To analyze the growth trends for the boys in Paul's family and determine the correct option, we need to calculate the differences in height between each age period for each boy. The given heights in feet and inches need to be converted to inches first, then we will calculate the growth increments.

### Step-by-Step Calculation:

Let's convert the heights to inches. Each foot is 12 inches.

#### Paul
- Age 8: [tex]\(4'8'' \Rightarrow (4 \times 12) + 8 = 56 \)[/tex] inches
- Age 12: [tex]\(5'0'' \Rightarrow (5 \times 12) = 60 \)[/tex] inches
- Age 16: [tex]\(5'11'' \Rightarrow (5 \times 12) + 11 = 71 \)[/tex] inches
- Age 20: [tex]\(6'1'' \Rightarrow (6 \times 12) + 1 = 73 \)[/tex] inches
- Age 24: [tex]\(6'1'' \Rightarrow (6 \times 12) + 1 = 73 \)[/tex] inches

Growth (in inches):
- 8 to 12: [tex]\(60 - 56 = 4\)[/tex]
- 12 to 16: [tex]\(71 - 60 = 11\)[/tex]
- 16 to 20: [tex]\(73 - 71 = 2\)[/tex]
- 20 to 24: [tex]\(73 - 73 = 0\)[/tex]

#### Bruce
- Age 8: [tex]\(4'6'' \Rightarrow (4 \times 12) + 6 = 54 \)[/tex] inches
- Age 12: [tex]\(4'11'' \Rightarrow (4 \times 12) + 11 = 59 \)[/tex] inches
- Age 16: [tex]\(5'10'' \Rightarrow (5 \times 12) + 10 = 70 \)[/tex] inches
- Age 20: [tex]\(6'1'' \Rightarrow (6 \times 12) + 1 = 73 \)[/tex] inches
- Age 24: [tex]\(6'1'' \Rightarrow (6 \times 12) + 1 = 73 \)[/tex] inches

Growth (in inches):
- 8 to 12: [tex]\(59 - 54 = 5\)[/tex]
- 12 to 16: [tex]\(70 - 59 = 11\)[/tex]
- 16 to 20: [tex]\(73 - 70 = 3\)[/tex]
- 20 to 24: [tex]\(73 - 73 = 0\)[/tex]

#### Tom
- Age 8: [tex]\(4'9'' \Rightarrow (4 \times 12) + 9 = 57 \)[/tex] inches
- Age 12: [tex]\(5'0'' \Rightarrow (5 \times 12) = 60 \)[/tex] inches
- Age 16: [tex]\(6'0'' \Rightarrow (6 \times 12) = 72 \)[/tex] inches
- Age 20: [tex]\(6'2'' \Rightarrow (6 \times 12) + 2 = 74 \)[/tex] inches
- Age 24: [tex]\(6'2'' \Rightarrow (6 \times 12) + 2 = 74 \)[/tex] inches

Growth (in inches):
- 8 to 12: [tex]\(60 - 57 = 3\)[/tex]
- 12 to 16: [tex]\(72 - 60 = 12\)[/tex]
- 16 to 20: [tex]\(74 - 72 = 2\)[/tex]
- 20 to 24: [tex]\(74 - 74 = 0\)[/tex]

#### Steve
- Age 8: [tex]\(4'7'' \Rightarrow (4 \times 12) + 7 = 55 \)[/tex] inches
- Age 12: [tex]\(4'11'' \Rightarrow (4 \times 12) + 11 = 59 \)[/tex] inches
- Age 16: [tex]\(5'11'' \Rightarrow (5 \times 12) + 11 = 71 \)[/tex] inches
- Age 20: [tex]\(6'1'' \Rightarrow (6 \times 12) + 1 = 73 \)[/tex] inches
- Age 24: [tex]\(6'1'' \Rightarrow (6 \times 12) + 1 = 73 \)[/tex] inches

Growth (in inches):
- 8 to 12: [tex]\(59 - 55 = 4\)[/tex]
- 12 to 16: [tex]\(71 - 59 = 12\)[/tex]
- 16 to 20: [tex]\(73 - 71 = 2\)[/tex]
- 20 to 24: [tex]\(73 - 73 = 0\)[/tex]

### Calculating Average Growth for Each Period
- Between ages 8 to 12: [tex]\((4, 5, 3, 4) \Rightarrow \frac{4 + 5 + 3 + 4}{4} = 4.0\)[/tex]
- Between ages 12 to 16: [tex]\((11, 11, 12, 12) \Rightarrow \frac{11 + 11 + 12 + 12}{4} = 11.5\)[/tex]
- Between ages 16 to 20: [tex]\((2, 3, 2, 2) \Rightarrow \frac{2 + 3 + 2 + 2}{4} = 2.25\)[/tex]
- Between ages 20 to 24: [tex]\((0, 0, 0, 0) \Rightarrow \frac{0 + 0 + 0 + 0}{4} = 0.0\)[/tex]

### Conclusion
Examining the average growth calculations:
- The largest average growth occurs between 12 to 16 years (average = 11.5 inches).

Therefore, the correct conclusion about the growth trend is:
c. Between the ages of 12 and 16, all four boys had the biggest jump in height.

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