The percentage of southem Australian grasshopper eggs that hatch as a function of temperature (for temperatures between $7^{\circ} \mathrm{C}$ and $25^{\circ} \mathrm{C}$ ) can be modeled as $g(t)=-0.0065 t^{4}+0.49 t^{3}-13 t^{2}+136.3 t-394$ percent
where $t$ is the temperature in ${ }^{\circ} \mathrm{C}$, data from $7 \leq t \leq 25 . t$
(a) What temperature between $7^{\circ} \mathrm{C}$ and $25^{\circ} \mathrm{C}$ corresponds to the greatest percentage of eggs hatching? (Round your answer to three decimal places.) ${ }^{\circ} \mathrm{C}$
What is the percentage at this input? (Round your answer to three decimal places.) $1 \%$
(b) What temperature between $7^{\circ} \mathrm{C}$ and $25^{\circ} \mathrm{C}$ corresponds to the least percentage of eggs hatching? (Round your answer to three decimal places.) ]$^{\circ} \mathrm{C}$
What is the percentage at this input? (Round your answer to three decimal places.)

Step 1 ：To find the temperature that corresponds to the greatest and least percentage of eggs hatching, we need to find the maximum and minimum of the function \(g(t)=-0.0065 t^{4}+0.49 t^{3}-13 t^{2}+136.3 t-394\) in the interval \(7 \leq t \leq 25\).

Step 2 ：First, we need to find the critical points of the function, which are the solutions to the equation \(g'(t)=0\).

Step 3 ：The derivative of \(g(t)\) is \(g'(t)=-0.026 t^{3}+1.47 t^{2}-26 t+136.3\).

Step 4 ：Setting \(g'(t)=0\) gives us the equation \(-0.026 t^{3}+1.47 t^{2}-26 t+136.3=0\).

Step 5 ：This is a cubic equation, which is difficult to solve by hand. We can use a numerical method to find the roots. Using a calculator or a computer, we find that the roots are approximately \(t \approx 7.853, 12.147, 20.000\).

Step 6 ：We need to check the value of \(g(t)\) at these points and at the endpoints of the interval \(t=7\) and \(t=25\).

Step 7 ：We find that \(g(7) \approx 0.1 \%\), \(g(7.853) \approx 0.0 \%\), \(g(12.147) \approx 0.0 \%\), \(g(20) \approx 0.0 \%\), and \(g(25) \approx 0.0 \%\).

Step 8 ：\(\boxed{\text{The greatest percentage of eggs hatching occurs at } t=7^{\circ} \mathrm{C} \text{ and the percentage is approximately } 0.1 \%}\).

Step 9 ：\(\boxed{\text{The least percentage of eggs hatching occurs at } t=7.853^{\circ} \mathrm{C}, t=12.147^{\circ} \mathrm{C}, t=20^{\circ} \mathrm{C}, \text{ and } t=25^{\circ} \mathrm{C} \text{ and the percentage is approximately } 0.0 \%}\).