A box with a square base and open top must have a volume of 78732 cm, We wish to find the dimensions of the box that minimize the amount of material used. First, find a formula for the surface area of the box in terms of only æ, the length of one side of the square base. [Hint: use the volume formula to express the height of the box in terms of r.] Simplify your formula as much as possible. A(z) = Next, find the derivative, A'(x). A' (x) = %3! Now, calculate when the derivative equals zero, that is, when A (z) 0. [Hint: multiply both sides by .) %3D A'(x) = 0 when = %3D We next have to make sure that this value of x gives a minimum value for the surface area. Let's use the second derivative test. Find A"(x). A"(x) = %3D Evaluate A"(x) at the r-value you gave above. NOTE: Since your last answer is positive, this means that the graph of A(x) is concave up around that value, so the zero of A'(x) must indicate a local minimum for A(). (Your boss is happy now.)
A box with a square base and open top must have a volume of 78732 cm, We wish to find the dimensions of the box that minimize the amount of material used. First, find a formula for the surface area of the box in terms of only æ, the length of one side of the square base. [Hint: use the volume formula to express the height of the box in terms of r.] Simplify your formula as much as possible. A(z) = Next, find the derivative, A'(x). A' (x) = %3! Now, calculate when the derivative equals zero, that is, when A (z) 0. [Hint: multiply both sides by .) %3D A'(x) = 0 when = %3D We next have to make sure that this value of x gives a minimum value for the surface area. Let's use the second derivative test. Find A"(x). A"(x) = %3D Evaluate A"(x) at the r-value you gave above. NOTE: Since your last answer is positive, this means that the graph of A(x) is concave up around that value, so the zero of A'(x) must indicate a local minimum for A(). (Your boss is happy now.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![A box with a square base and open top must have a volume of 78732 cm. We wish to find the dimensions of the box that
minimize the amount of material used.
First, find a formula for the surface area of the box in terms of only æ, the length of one side of the square base.
[Hint: use the volume formula to express the height of the box in terms of r.]
Simplify your formula as much as possible.
A(z) =
Next, find the derivative, A'(x).
A' (x) =
Now, calculate when the derivative equals zero, that is, when A (2)
0. [Hint: multiply both sides by a)
= 0 when I =
We next have to make sure that this value of z gives a minimum value for the surface area. Let's use the second derivative
test. Find A"(x).
A"(x) =
Evaluate A"(x) at the x-value you gave above.
NOTE: Since your last answer is positive, this means that the graph of A(x) is concave up around that value, so the
zero of A'(x) must indicate a local minimum for A(x). (Your boss is happy now.)
Submit Question](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F495a786d-1fb3-471c-b84a-fc25099ed51b%2F987f56bb-670a-43dd-85ff-1b63136a2b18%2Fe7h2jjmn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A box with a square base and open top must have a volume of 78732 cm. We wish to find the dimensions of the box that
minimize the amount of material used.
First, find a formula for the surface area of the box in terms of only æ, the length of one side of the square base.
[Hint: use the volume formula to express the height of the box in terms of r.]
Simplify your formula as much as possible.
A(z) =
Next, find the derivative, A'(x).
A' (x) =
Now, calculate when the derivative equals zero, that is, when A (2)
0. [Hint: multiply both sides by a)
= 0 when I =
We next have to make sure that this value of z gives a minimum value for the surface area. Let's use the second derivative
test. Find A"(x).
A"(x) =
Evaluate A"(x) at the x-value you gave above.
NOTE: Since your last answer is positive, this means that the graph of A(x) is concave up around that value, so the
zero of A'(x) must indicate a local minimum for A(x). (Your boss is happy now.)
Submit Question
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 7 steps with 1 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
