A box without a top is to be made from a rectangular piece of cardboard, with dimensions 9 in. by 10 in., by cutting out square corners with side length x and folding up the sides.
What is the optimal volume of the box, and what is the optimal cut?

The optimal volume is 63.1142 in3, with a cut of 1.5767 inches

Step-by-step explanation:

If we make a cut of x inches to create the box, the dimensions of the box will be:

Length = 10 - 2x

Width = 9 - 2x

Height = x

So the volume of the box would be:

Volume = (10 - 2x) * (9 - 2x) * x

Volume = 4x3 - 38x2 + 90x

To find the maximum volume, we need to take the derivative of the volume and find the values of x where it is equal to zero:

dV/dx = 12x2 - 76x + 90 = 0

6x2 - 38x + 45 = 0

Using Bhaskara's formula, we have:

Delta = 38^2 - 4*6*45 = 364

sqrt(Delta) = 19.08

x1 = (38 + 19.08) / 12 = 4.7567

x2 = (38 - 19.08) / 12 = 1.5767

Testing these values in the volume equation, we have:

Volume1 = 4*(4.7567)^3 - 38*(4.7567)^2 + 90*(4.7567) = -1.1883

(Negative value for the volume is not valid)

Volume1 = 4*(1.5767)^3 - 38*(1.5767)^2 + 90*(1.5767) = 63.1142

So the optimal volume is 63.1142 in3, with a cut of 1.5767 inches

(0,-3) see attached graph

step-by-step explanation:

a. to solve the system, graph the two equations.

x = y+3 becomes y=x-3 has m=1/1 and b=-3. start at (0,-3) on the y-axis and plot a point. then move up 1 unit and 1 unit to the right. plot a point and connect.

y=-4x+3 has m =-4/1 and b=-3. start at (0,-3) on the y-axis and plot a point. then move down four units and to the right 1 unit. plot a point and connect.

the solution is the point (x,y) where the two lines intersect. by the graph below it is (0,-3).

b. to solve by substitution, substitute y = -4x-3 into x= y+3

x= (-4x-3) +3

x=-4x

5x=0

x=0

now substitute x = 0 back in to find y.

y= -4(0)-3

y=0-3

y=-3

solution is (0,-3)

c. use elimination by adding the equations together.

x-y=3

4x+y=-3

5x=0

x=0

then substitute into one equation to find y.

y=-4(0)-3

y=-3

7.25

step-by-step explanation:

we know that 7 equals to 7.0. we also know that 1/4=0.25. so we can add the two together and we get 7.25