If 300 cm2 of material is available to make a box with a square base and an open top, find the maximum volume of the box in cubic centimeters. Answer to the nearest cubic centimeter without commas. For example, if the answer is 2,000 write 2000
Answers
Answer:
500 cc.
Step-by-step explanation:
We can write 2 equations for the above problem-one based on area of material to be used, other based on volume of formed structure.
Assuming V to be volume, a to be height and b to be base length of cuboid, the equations are:
V = a x b^2
300 = b^2 + (4 x a x b)
Next, substituting the value of a from the first eqn into the second eqn gives you a relation between b and V. Now, differentiating w.r.t. b for maximisation, we get the value of b to be 10 cms.
Substituting in the remaining equations, we calculate a to be 5 cms and V to be 500 cc.
Answer:
500 cm³
Step-by-step explanation:
Let the length of the square be L
Let the Height be H
Let the volume be V
Volume = Length x Breadth x Height
V= L²H
H = V/L²
Surface area of an open square based box = Length² + 4Length x Height
L² + 4LH = 300
H = V/L² -------------------------- [ 1 ]
L² + 4LH = 300 ----------------- [ 2 ]
Sub [ 1 ] into [ 2 ]:
L² + 4L(V/L²) = 300
L² + 4V/L = 300
L³ + 4V = 300L
L³ - 300L + 4V = 0
4V = 300L - L³
V = 75L - 1/4 L³
Differentiate:
dv/dL = 75 - 3/4 L²
Find max value of L:
75 - 3/4 L² = 0
3/4 L² = 75
L² = 100
L = 10 cm
Find Height:
L² + 4LH = 300
10² + 4(10)H = 300
100 + 40H = 300
40H = 200
H = 5 cm
Find maximum volume:
Volume = Length x Length x Height
Volume = 10 x 10 x 5 = 500 cm³
Answer: The maximum volume is 500 cm³