3. The radioactive substance lodine- 131 is used in medicine to measure liver and heart activity and in the treatment of certain cancers. It has a half- life of eight days, \( \mathrm{i} . \mathrm{e} \). the quantity of lodine-131 decays by a factor of 2 in eight days. A hospital purchases 30 g of the substance. In eight days 15 g remain. The decay can be modelled using the exponential model: \[ Q=a e^{b t} \] where \( a \) and \( b \) are constants, \( t \) is time measured in days and \( Q \) is the quantity of the substance. (i) Find the value of each of the constants \( a \) and \( b \). (ii) What quantity of the substance will remain after 20 days? (iii) Find the time taken for the substance to decay to 1.49 g .

Suppose \( \int_{2}^{4} f(x) d x=6, \int_{2}^{7} f(x) d x=-2 \), and \( \int_{2}^{7} g(x) d x=4 \). Evaluate the following integrals. \( \int_{7}^{2} g(x) d x=-4 \) (Simplify your answer.) \( \int_{2}^{7} 5 g(x) d x=\square \) (Simplify your answer.)

Suppose \( \int_{2}^{4} f(x) d x=6, \int_{2}^{7} f(x) d x=-2 \), and \( \int_{2}^{7} g(x) d x=4 \). Evaluate the following integrals. \( \int_{7}^{2} g(x) d x=-4 \) (Simplify your answer.) \( \int_{7}^{7} 5 g(x) d x=20 \) 2 (Simplify your answer.) \( \int_{7}^{7}[g(x)-f(x)] d x=\square \) 2 (Simplify your answer.)

\( \int_{2}^{7} 5 g(x) d x=\square \) (Simplify your answer.)

What is the average rate of change on the function \( f(x)=x^{2} \) between \( x= \) 1 and \( x=3 \) ? -1

Suppose \( \int_{2}^{4} f(x) d x=6, \int_{2}^{7} f(x) d x=-2 \), and \( \int_{2}^{7} g(x) d x=4 \). Evaluate the following integrals. (Simplify your answer.) 7 \( \int_{2} 5 g(x) d x=20 \) (Simplify your answer.) 7 \( \int[g(x)-f(x)] d x=6 \) 2 \( (S i m p l i f y \) 7 \( \int_{2}^{7}[5 g(x)-f(x)] d x=\square \) 2