The bacteria in a Petri dish culture are self-duplicating at a rapid pace. The relationship between the elapsed time t, in minutes, and the number of bacteria, B(t), in the Petri dish is modeled by the following function. B(t)=100*2^((t)/( 10)) How many minutes will it take for the culture to achieve 10,000 bacteria? Round your answer, if necessary, to the nearest hundredth. minutes

Question

The bacteria in a Petri dish culture are self-duplicating at a rapid pace. The relationship between the elapsed time  t, in minutes, and the number of bacteria,  B(t), in the Petri dish is modeled by the following function.  B(t)=100*2^((t)/( 10)) How many minutes will it take for the culture to achieve 10,000 bacteria? Round your answer, if necessary, to the nearest hundredth. minutes

Bytelearn · Accepted Answer

Given Function: We are given the function B(t) = 100 * 2^(t/10) and we need to find the time t when B(t) = 10,000. We can set up the equation 10,000 = 100 * 2^(t/10) and solve for t.Isolate Exponential Term: Divide both sides of the equation by 100 to isolate the exponential term on one side. This gives us 10,000 / 100 = 2^(t/10).Simplify Equation: Simplify the left side of the equation to get 100 = 2^(t/10).Take Logarithm: To solve for t, we need to take the logarithm of both sides. We will use the base 2 logarithm because the base of the exponent is 2. This gives us log2(100) = log2(2^(t/10)).Calculate Logarithm: Using the property of logarithms that log_b(b^x) = x, we can simplify the right side to get log2(100) = t/10.Solve for t: Now we need to calculate log2(100). Since 2^6 = 64 and 2^7 = 128, log2(100) is between 6 and 7. We can use a calculator to find the exact value.Round Final Answer: Using a calculator, we find that log2(100) is approximately 6.643856. So we have 6.643856 = t/10.Multiply both sides of the equation by 10 to solve for t. This gives us 10 * 6.643856 = t.Calculate the value of t by multiplying 6.643856 by 10 to get t ≈ 66.43856.Round the answer to the nearest hundredth as requested. The rounded value of t is 66.44 minutes.

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