Answer :
To determine the values of \(a\) and \(b\) for the exponential function \(f(x) = ab^x\) passing through the points \((0, 11000)\) and \((3, 704)\), we can follow these steps:
1. Using the point (0, 11000):
When \(x = 0\), \(f(x) = a b^0 = a\). So, from the point \((0, 11000)\),
[tex]\[ f(0) = a = 11000. \][/tex]
Therefore,
[tex]\[ a = 11000. \][/tex]
2. Using the point (3, 704):
When \(x = 3\), \(f(x) = ab^3\). So, from the point \((3, 704)\),
[tex]\[ f(3) = 11000b^3 = 704. \][/tex]
3. Solve for \(b\):
We set up the equation from the above point:
[tex]\[ 11000b^3 = 704. \][/tex]
To isolate \(b^3\), divide both sides by 11000:
[tex]\[ b^3 = \frac{704}{11000}. \][/tex]
Simplifying the fraction:
[tex]\[ b^3 = \frac{704}{11000} = \frac{704 \div 16}{11000 \div 16} = \frac{44}{687.5}. \][/tex]
Further simplifying:
[tex]\[ b^3 = \frac{44 \div 2}{687.5 \div 2} = \frac{22}{343.75}. \][/tex]
To simplify more accurately in common fraction or decimal:
[tex]\[ b^3 = 0.064. \][/tex]
4. Find the cube root of \(0.064\):
[tex]\[ b = \sqrt[3]{0.064}. \][/tex]
Calculating the cube root:
[tex]\[ b = 0.4. \][/tex]
Now we have the values for \(a\) and \(b\):
[tex]\[ a = 11000, \][/tex]
[tex]\[ b = 0.4. \][/tex]
Thus, the values of \(a\) and \(b\) are:
[tex]\[ \begin{array}{l} a = 11000 \\ b = 0.4 \end{array} \][/tex]
1. Using the point (0, 11000):
When \(x = 0\), \(f(x) = a b^0 = a\). So, from the point \((0, 11000)\),
[tex]\[ f(0) = a = 11000. \][/tex]
Therefore,
[tex]\[ a = 11000. \][/tex]
2. Using the point (3, 704):
When \(x = 3\), \(f(x) = ab^3\). So, from the point \((3, 704)\),
[tex]\[ f(3) = 11000b^3 = 704. \][/tex]
3. Solve for \(b\):
We set up the equation from the above point:
[tex]\[ 11000b^3 = 704. \][/tex]
To isolate \(b^3\), divide both sides by 11000:
[tex]\[ b^3 = \frac{704}{11000}. \][/tex]
Simplifying the fraction:
[tex]\[ b^3 = \frac{704}{11000} = \frac{704 \div 16}{11000 \div 16} = \frac{44}{687.5}. \][/tex]
Further simplifying:
[tex]\[ b^3 = \frac{44 \div 2}{687.5 \div 2} = \frac{22}{343.75}. \][/tex]
To simplify more accurately in common fraction or decimal:
[tex]\[ b^3 = 0.064. \][/tex]
4. Find the cube root of \(0.064\):
[tex]\[ b = \sqrt[3]{0.064}. \][/tex]
Calculating the cube root:
[tex]\[ b = 0.4. \][/tex]
Now we have the values for \(a\) and \(b\):
[tex]\[ a = 11000, \][/tex]
[tex]\[ b = 0.4. \][/tex]
Thus, the values of \(a\) and \(b\) are:
[tex]\[ \begin{array}{l} a = 11000 \\ b = 0.4 \end{array} \][/tex]
