What are you given variables? Surface area of box: 10800cm² Volume of box: s²h <span>Where: </span>SA=surface area s = side of square base h = height of box
Make the height of the box in terms of s You can write the formula for the surface area of the box in terms of s and h like so: S.A. = s² + 4sh Where: S.A. = surface area or 1200 cm², s² = the square base, and 4sh = the four 'walls' of the box. 10800 = s² + 4sh 10800 - s² = 4sh (10800 - s²)/(4s) = h
Substitute h (in terms of s) into the formula for volume. v(s) = s²((10800- s²)/(4s)) // Simplify. v(s) = s(10800 - s²)/4 // Expand. v(s) = 2700s - (1/4)s^3 // To find the largest possible volume of the box, you find the maximum value of this function.
Take the derivative of the volume function using the Power Theorem. v'(s) = 2700 - (3/4)s² // Zeroes of the d/dx v function will give you the x values that correspond to local extrema in the v function. 0 = 2700 - (3/4)s² // Solve for zeroes. -2700= (-3/4)s² 3600= s² Your two zero values for s are -60 and 60.
Take the second derivative to see if the s values will give you a local maximum or minimum. v"(s) = -(3/2)s v"(-60) = -(3/2)(-60) v"(-60) = 90 // This indicates a local minimum for v at s, not what we are looking for. v"(60) = -(3/2)(60) v"(60)
= -90 // This indicates a local maximum for v at s, which is what we
are looking for, the maximum volume of the box. This makes sense because
if you remember what we assigned the variable s to, a side length, side
lengths cannot be negative.
Once we have found the value for s, we can substitute it into the function we created for the volume of the box. v(s) = 2700s- (1/4)s^3 v(s) = 2700(60) - (1/4)(60)^3 v(s) = 162000- (1/4)(216000) v = 162000- 54000
The largest possible volume of the box is 108000 cubic centimeters.
The colonists agitation led to various acts of rebellion, a moajor one being the Boston Tea Party, where colonists dumped hundreds of boxes of tea into Boston Harbor. This caused more tension by making the British authorities tighten their control, angering the colonists even more, eventually leading to the American Revolution.
B. The new technologies would reduce positive checks on population.
Explanation:
Since there would be better and more efficient methods of farming that would reduce the positive checks that happen to the population. The positive checks would at first be a lot less impactful than before the technology. Disease and natural disasters would not cripple a population until it grows to a much larger number.
