Answer:
The number of bacteria in the dish after 9.9 minutes is 478
Step-by-step explanation:
The form of the function is [tex]f(x)=a.b^{x}[/tex] , where a is the initial amount (value f(x) at x = 0)
To find a and b let us use two points from the graph and substitute their coordinates in the equation
∵ The graph passes through point (1 , 1)
∴ x = 1 and y = 1
- Substitute them in the equation
∵ [tex]1=a.b^{1}[/tex]
∴ a.b = 1 ⇒ (1)
∵ The graph passes through point (1 , 1)
∴ x = 3 and y = 4
- Substitute them in the equation
∵ [tex]4=a.b^{3}[/tex]
∴ a.b³ = 4 ⇒ (2)
Use equation (1) to find a in terms of b
∵ a.b = 1
- Divide both sides by b
∴ [tex]a=\frac{1}{b}[/tex] ⇒ (3)
- Substitute a in equation (2) by equation (3)
∴ [tex]\frac{1}{b}[/tex] . b³ = 4
- Remember the rule [tex]\frac{a^{m}}{a^{n}}=a^{m-n}[/tex]
∵ [tex]\frac{1}{b}[/tex] . b³ = [tex]b^{3-1}[/tex] = b²
∴ b² = 4
- Take √ for both sides
∴ b = 2
Substitute the value of b in equation (3) to find a
∵ a = [tex]\frac{1}{2}[/tex]
∴ a = 0.5
∴ [tex]f(x)=0.5(2)^{x}[/tex]
∵ f(x) represents the number of bacteria after x minutes
∵ x = 9.9 minutes
- Substitute x by 9.9 to find the number of bacteria
∵ [tex]f(9.9)=0.5(2)^{9.9}[/tex]
∴ f(9.9) = 477.7128917
- Round it to the nearest whole number
∴ f(9.9) = 478
∴ The number of bacteria in the dish after 9.9 minutes is 478
