7) An open box is to be made from a 17 inch by 32 inch rectangular piece of card

Question

7) An open box is to be made from a 17 inch by 32 inch rectangular piece of cardboard by cutting equal squares from the corners and turning up the sides Find the volume of the lar est box that can be made in this manner. % Notes: Use the "Solve" feature of your calculator Maximiz Do not do any rounding Be sure to include units. Be sure to state the domain of your function. In the interest of timeou nat have to prove that your answer is, in fact, the absolute maximum. 32 in 1-2 x 7in V(x) = (o 4 e (0, 8.5)

Explanation / Answer

let x be the length of sides of small square that was cut from sides ,domain of x is (0,8.5)

volume of box v=(32-2x)*(17-2x)*x

v=544x -98x2+4x3

form maximum volume dv/dx =0 ,d2v/dx2 <0

dv/dx =544 -98*2x+4*3x2=0

12x2-196x+544=0

x=[196+[1962-(4*12*544)]]/(2*12),x=[196+[1962-(4*12*544)]]/(2*12)

x=[49+769]/6,x=[49-769]/6

x =3.5449,x=12.788

therefore x =3.5449

d2v/dx2 =6x-98

at x =3.5449

d2v/dx2 =6*3.5449 -98 =-76.7306<0

so maximum volume at x =3.5449

maximum volume =(32-3.5449)*(17-3.5449)*3.5449

maximum volume =1357.2

volume of largest box =1357.2in3

Related Questions

Navigate

• Browse (All)
• Browse 7
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.