Hiwatha Manufacturing has two factories that produce three grades of paper: low grade, medium grade, and high grade. It needs to supply at least 42 tons of low grade, 30 tons of medium grade, and
12 tons of high grade paper. Factory A produces 3 tons of low grade, 1 ton of medium grade, and 3
tons of high grade paper daily, and costs $1200 per day to operate. Factory B produces 6 tons of low
grade, 6 ton of medium grade, 1 tons of high grade paper daily, and costs $1800 per day to operate.
How many days should each factory operate to fill the orders at minimum cost?

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4

5

step-by-step explanation: