Question

The graph shown represents the number of bacteria in a dish after a certain number of minutes. It is a function in the form f(x) = a · bx.

Answer 1

Answer:

The number of bacteria in the dish after 9.9 minutes is 478

Step-by-step explanation:

The form of the function is $f(x)=a.b^{x}$ , 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

∵ $1=a.b^{1}$

∴ a.b = 1 ⇒ (1)

∵ The graph passes through point (1 , 1)

∴ x = 3 and y = 4

- Substitute them in the equation

∵ $4=a.b^{3}$

∴ a.b³ = 4 ⇒ (2)

Use equation (1) to find a in terms of b

∵ a.b = 1

- Divide both sides by b

∴ $a=\frac{1}{b}$ ⇒ (3)

- Substitute a in equation (2) by equation (3)

∴ $\frac{1}{b}$ . b³ = 4

- Remember the rule $\frac{a^{m}}{a^{n}}=a^{m-n}$

∵ $\frac{1}{b}$ . b³ = $b^{3-1}$ = b²

∴ b² = 4

- Take √  for both sides

∴ b = 2

Substitute the value of b in equation (3) to find a

∵ a = $\frac{1}{2}$

∴ a = 0.5

∴ $f(x)=0.5(2)^{x}$

∵ 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

∵ $f(9.9)=0.5(2)^{9.9}$

∴ 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