at the beginning of the year, a blogger has 325 subscribers. since then, the number of subscribers to her…

at the beginning of the year, a blogger has 325 subscribers. since then, the number of subscribers to her blog has been doubling every 4 months.\nby what factor is the number of subscribers increasing every month?\na (sqrt4{325}) or (325^{\frac{1}{4}})\n b (sqrt{2}) or (2^{\frac{1}{2}})\n c (sqrt4{2}) or (2^{\frac{1}{4}})\n d (sqrt{4}) or (4^{\frac{1}{2}})

at the beginning of the year, a blogger has 325 subscribers. since then, the number of subscribers to her blog has been doubling every 4 months.\nby what factor is the number of subscribers increasing every month?\na (sqrt4{325}) or (325^{\frac{1}{4}})\n b (sqrt{2}) or (2^{\frac{1}{2}})\n c (sqrt4{2}) or (2^{\frac{1}{4}})\n d (sqrt{4}) or (4^{\frac{1}{2}})

Answer

Explanation:

Step1: Define the exponential growth model

Let ( N(t) ) be the number of subscribers at time ( t ) (in months), with the initial number ( N(0) = 325 ). The growth is doubling every 4 months, so we can write the formula as ( N(t)=325\times2^{\frac{t}{4}} ), where the exponent ( \frac{t}{4} ) represents the number of 4 - month periods in ( t ) months.

Step2: Find the monthly growth factor

We can rewrite the function ( N(t) ) in the form ( N(t)=325\times a^{t} ), where ( a ) is the monthly growth factor. Comparing ( 325\times2^{\frac{t}{4}} ) with ( 325\times a^{t} ), we have ( a^{t}=2^{\frac{t}{4}} ). By the property of exponents, if ( a^{t}=b^{t} ) for all ( t ), then ( a = b ) (when ( t\neq0 )). So we can rewrite ( 2^{\frac{t}{4}} ) as ( (2^{\frac{1}{4}})^{t} ). Using the radical - exponent relationship ( a^{\frac{m}{n}}=\sqrt[n]{a^{m}} ), ( 2^{\frac{1}{4}}=\sqrt[4]{2} ). So the monthly growth factor ( a = \sqrt[4]{2}=2^{\frac{1}{4}} ).

Answer:

C. (\sqrt[4]{2}) or (2^{\frac{1}{4}})