The percentage of patients $$ P $$ who have survived $$ t $$ years after initial diagnosis of a certain disease is modeled by the function $$ P(t)=100(0.7)^{t} $$. (a) According to the model, what percent of patients survive 1 year after initial diagnosis? (b) What percent of patients survive 3 years after initial diagnosis? (c) Explain the meaning of the base 0.7 in the context of this problem. (a) According to the model, $$ \square \% $$ of patients survive 1 year after initial diagnosis. (Type an integer or a decimal.)

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UpStudy AI · Accepted Answer

1. The given model is 
   $$
   P(t) = 100(0.7)^t.
   $$

2. For $$ t = 1 $$, substitute into the formula:
   $$
   P(1) = 100(0.7)^1 = 100 \times 0.7 = 70.
   $$

3. Therefore, $$ \boxed{70} \% $$ of patients survive 1 year after the initial diagnosis.
70% of patients survive 1 year after initial diagnosis.

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