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MCR3U - OPTIONAL ASSIGNMENT
Part A : choose one

A culture has 570 bacteria. The number of bacteria doubles every
12h. How long will it take to have 2152 bacteria in the culture?

2x^(2)-3x

MCR $3$ U - OPTIONAL ASSIGNMENT $\newline$ Part A : choose one $\newline$ $1$ . A culture has $570$ bacteria. The number of bacteria doubles every $12 \mathrm{~h}$ . How long will it take to have $2152$ bacteria in the culture? $\newline$ $2 x^{2}-3 x$

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Which word problem matches the one-step equation?

Full solution

Q. MCR

3

U - OPTIONAL ASSIGNMENT

\newline

Part A : choose one

\newline

1

. A culture has

570

bacteria. The number of bacteria doubles every

12 \mathrm{~h}

. How long will it take to have

2152

bacteria in the culture?

\newline

2 x^{2}-3 x

Denote Initial Bacteria Number: Let's denote the initial number of bacteria as $B_0$ and the number of bacteria after $t$ hours as $B(t)$ . The formula for exponential growth is $B(t) = B_0 \times 2^{(t/T)}$ , where $T$ is the doubling time in hours.
Set Up Exponential Growth Formula: We know $B_0 = 570$ , $B(t) = 2152$ , and $T = 12$ . We need to solve for $t$ . So, we set up the equation $2152 = 570 \times 2^{t/12}$ .
Isolate Exponential Part: To isolate the exponential part, we divide both sides by $570$ : $2152 / 570 = 2^{t/12}$ .
Calculate Left Side: Calculating the left side gives us $2152 / 570 = 3.77$ (rounded to two decimal places).
Take Natural Logarithm: Now we need to solve for $t$ in the equation $3.77 = 2^{\frac{t}{12}}$ . We can do this by taking the logarithm of both sides. Let's use the natural logarithm (ln): $\ln(3.77) = \ln(2^{\frac{t}{12}})$ .
Rewrite Equation with Logarithms: Using the property of logarithms that $\ln(a^b) = b\cdot\ln(a)$ , we can rewrite the equation as $\ln(3.77) = \left(\frac{t}{12}\right)\cdot\ln(2)$ .
Solve for t: Now we solve for t: $t = (\ln(3.77) / \ln(2)) \times 12$ .
Perform Calculation: Calculating the right side gives us $t = (\ln(3.77) / \ln(2)) \times 12 = (1.327 / 0.693) \times 12$ .
Round Up to Nearest Multiple: After performing the calculation, we get $t = (1.327 / 0.693) \times 12 = 22.98$ hours. However, since the bacteria can only double at $12$ -hour intervals, we need to round up to the nearest multiple of $12$ . So, $t = 24$ hours.

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