Question

in 1950, computers were able to perform 1 calculation per second. as technology progressed, the calculation rate increased by 58.49% each year.
select the equation that can be used to find the number of years it took for computers to perform (10^{10}) calculations per second. how many years did it take for computers to calculate at that rate?
a. (1 = 10^{10}(1.5849)^t)
it took about 25 years to increase the speed to (10^{10}) calculations per second.
b. (10^{10} = 1(1.15849)^t)
it took about 50 years to increase the speed to (10^{10}) calculations per second.
c. (10^{10} = 1(58.49)^t)
it took about 6 years to increase the speed to (10^{10}) calculations per second.
d. (10^{10} = (0.5849)^t)
it took about 43 years to increase the speed to (10^{10}) calculations per second.

Explanation:

Step1: Identify growth formula

The exponential growth formula is $A = P(1 + r)^t$, where $P=1$ (initial calculations), $r=0.5849$ (58.49% growth), $A=10^{10}$. Substitute values:
$$10^{10} = 1(1.5849)^t$$

Step2: Solve for t using logarithms

Take natural log of both sides:
$$\ln(10^{10}) = \ln(1.5849^t)$$
Use log power rule $\ln(a^b)=b\ln(a)$:
$$10\ln(10) = t\ln(1.5849)$$
Solve for $t$:
$$t = \frac{10\ln(10)}{\ln(1.5849)}$$
Calculate values: $\ln(10)\approx2.3026$, $\ln(1.5849)\approx0.461$
$$t \approx \frac{10\times2.3026}{0.461} \approx 50$$

Answer:

B. $10^{10} = 1(1.5849)^t$
It took about 50 years to increase the speed to $10^{10}$ calculations per second.