Respuesta :
Answer:
A. [tex]a=350[/tex] and [tex]r=75\%[/tex]
B. 5744.6.
Step-by-step explanation:
We have been given an exponential function [tex]y=350(1+0.75)^t[/tex] and we are asked to identify the initial amount and the rate of growth of our given function.
A. Since we know that an exponential function is in form: [tex]y=a*b^x[/tex], where,
a = Initial value,
b = For growth b is in form (1+r), where r is in decimal form.
So, the exponential growth function is [tex]y=a*(1+r)^x[/tex]
Upon comparing the given exponential function with our exponential function we can see that a equals 350 and r equals 0.75.
Since growth rate is given in decimal form, so let us convert it as percent by multiplying by 100.
[tex]r=0.75*100=75\%[/tex]
Therefore, initial amount a is 350 and rate of growth is 75% for our given function.
B. Let us evaluate our function at t=5 by substituting t=5 in our given function.
[tex]y=350(1+0.75)^5[/tex]
[tex]y=350(1.75)^5[/tex]
[tex]y=350*16.4130859375[/tex]
[tex]y=5744.580078125\approx 5744.6[/tex]
Therefore, at t=5 we will get 5744.6.
We will see that the initial amount is A = 350, and the rate of growth is r = 75% per unit of time.
Finally, the function evaluated in t = 5 is equal to 5,744.6
A general exponential growth equation is written as:
f(t) = A*(1 + r)^t
Where:
A is the initial amount, such that:
f(0) = A.
r is the rate of growth, and t is the variable.
Here the given equation is:
y = 350*(1 + 0.75)^t
So we can see that the initial value is:
A = 350
And the rate is:
r = 0.75
But we want to write the rate in percent form, so we multiply it by 100%.
r = 0.75*100% = 75%.
Finally, we want to evaluate the function in t = 5, this gives:
y = 350*(1 + 0.75)^5 = 5,744.6
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