Answer:
a) The equation that represents how old Monique's son will be when he is 50 inches tall is [tex]a = 2 + \frac{8}{21}\cdot (h-34)[/tex].
b) Monique's son will be 8 years old when he is 50 inches tall.
Step-by-step explanation:
a) From statement we see that Monique's son grows at a constant rate and observes the following linear function:
[tex]h (t) = \dot h \cdot t + h_{o}[/tex] (Eq. 1)
Where:
[tex]h_{o}[/tex] - Initial height of Monique's son, measured in inches.
[tex]\dot h[/tex] - Growth rate, measured in inches per year.
[tex]t[/tex] - Time, measured in years.
The growth rate of the average boy is:
[tex]\dot h = 2\,\frac{5}{8}\,\frac{in}{yr}[/tex]
[tex]\dot h = \left(\frac{16}{8}+\frac{5}{8} \right)\,\frac{in}{yr}[/tex]
[tex]\dot h = \frac{21}{8} \,\frac{in}{yr}[/tex]
If we know that [tex]\dot h = \frac{21}{8} \,\frac{in}{yr}[/tex], [tex]t = 0\,yr[/tex] and [tex]h(t) = 34\,in[/tex], then the initial height of Monique's son is:
[tex]34\,in = \left(\frac{21}{8}\,\frac{in}{yr} \right)\cdot (0\,yr)+h_{o}[/tex]
[tex]h_{o} = 34\,in[/tex]
Then, the height of Monique's son as a function of age is represented by:
[tex]h(t) = \frac{21}{8}\cdot t +34[/tex] (Eq. 2)
The age of Monique's son ([tex]a[/tex]), expressed in years, is represented by the following formula:
[tex]a = 2+t[/tex] (Eq. 3)
Now we clear time within (Eq. 2):
[tex]h-34 = \frac{21}{8}\cdot t[/tex]
[tex]t = \frac{8}{21}\cdot (h-34)[/tex]
Therefore, the age of Monique's son is modelled after this:
[tex]a = 2 + \frac{8}{21}\cdot (h-34)[/tex] (Eq. 4)
b) If we know that [tex]h = 50\,in[/tex], then the age of Monique's son will be:
[tex]a = 2 + \frac{8}{21}\cdot (50-34)[/tex]
[tex]a = 8.095\,years.[/tex]
Monique's son will be 8 years old when he is 50 inches tall.
