Answer:
Volume = 5861.64 cubic inches
Explanation:
We were given that we should make an open rectangular box from a cardboard of dimension 25 inches by 49 inches
We are to cut congruent squares from the corners and folding up the sides
The box is open at the top
We will proceed to solve as shown below:
[tex]\begin{gathered} Length=49in \\ Width=25in \\ \text{Taking a unit of square from each side, we have:} \\ New\text{ }Length=49-2x;x<24.5 \\ New\text{ }Width=25-2x;x<12.5 \\ Height=x \\ \text{So, the doamin of this is \lparen0, 12.5\rparen} \\ V(x)=x(49-x)(25-x) \\ V(x)=x^3-74x^2+1225x \\ \text{Taking the derivative of both sides, we have:} \\ V^{\prime}(x)=3x^2-148x+1225 \\ Equate\text{ }V^{\prime}(x)\text{ to zero, we have:} \\ 3x^2-148x+1225=0 \\ \text{Solving using quadratic formula, we have:} \\ x=38.81273011,10.52060321 \\ Either\text{ of these two values is either the maximum or minimum} \end{gathered}[/tex]We will proceed as shown below:
[tex]\begin{gathered} x=38.81273011,10.52060321 \\ \text{Take a second derivative. The maximum point is the point where x returns a negative value, we have:} \\ V^{\prime}(x)=3x^2-148x+1225 \\ V^{^{\prime}}^{\prime}(x)=6x-148 \\ \text{We substitute both values into this, we have:} \\ when:x=38.81273011 \\ V^{\prime}^{\prime}^(x)=6(38.81273011)-148 \\ V^{\prime\prime}(x)=84.87638066 \\ when:x=10.52060321 \\ V^{\prime\prime}(x)=6(10.52060321)-148 \\ V^{\prime\prime}(x)=-84.87638074 \\ \text{Therefore, the value of the height that gives the greatest volume is 10.52060321 inches} \end{gathered}[/tex]Hence, the maximum volume is given by:
[tex]\begin{gathered} x=10.52060321 \\ V(x)=x^3-74x^2+1225 \\ V(x)=5861.64in^3 \end{gathered}[/tex]