Miguel is designing shipping boxes that are rectangular prisms. the shape of one box, with height h in feet, has a volume defined by the function v(h) = h(–h + 10)(–h + 8). what is the maximum volume for the domain 0 < h < 10? round to the nearest cubic foot.

105 cubic foot

Step-by-step explanation:

We have given with the function:-

V(h) = h(–h + 10)(–h + 8)

1) Let's simplify the function by applying distributive property to the right side

V(h) = h(–h + 10)(–h + 8)

       = {h(-h) + h(10)}(-h+8)

       = (-h²+10h)(-h+8)

       = (-h²)(-h)+(-h²)(8)+(10h)(-h)+(10h)(8)

       = -8h²-10h²+80h

V(h)   = - 18h² + 80h

2) Let's plugin the value of h from 1 to 9 (since 0<h<10) into the function V(h)   = - 18h² + 80h and create a table as shown below:-

h V(h)

163

296

3105          <-------------maximum volume

496

575

648

721

80

9  -9

Looking at the above table, we can conclude the maximum volume for the domain 0<h<10 is 105 cubic foot.

the graph in the attached figure

step-by-step explanation:

we have

using a graphing tool

the graph in the attached figure

the roots are ( x-intercepts) x=-0.5,x=1,x=3

the y-intercept is the point (0,3)

step-by-step explanation: