Given a flat piece of cardboard of 64 ft^2. What dimensions will yield an opened box with a square base of the largest volume?
The first step to solving this maximization problem is to understand the 64 ft^2 of cardboard is the constraint and surface area. The equation for surface area of the opened box here is 64 = x^2 + 4xy. Sketch and label the box to visualize. Recall what is to be maximized the volume of the opened box for which the formula is V = x*x*y. Making a substitution of y using the constraint, V = 16x - 0.25x^3. In differentiating the equation for Volume, setting it equal to 0 and solving for x, x = +/- (64/3)^0.5. Since dimensions cannot be negative, x=(64/3)^0.5 maximizes Volume. Plug x back into the constraint equation to solve for y; y=[64-(64/3)]/(4*(64/3)^0.5)
Find an equation of the line passing through points (0,0) and (15,-7).
The first step in finding an equation of a line is to find the slope. Recall the formula of slope is the difference in y divided by the difference in x. In this example, 15-0=15 and -7-0=-7. Therefore the slope is -15/7. The equation of a line in point-slope form is y-y1 = m(x-x1) where (x1,y1) is a point on the line. Let's say (0,0), which is given, is (x1,y1). The equation of the line is therefore y-0 = (-15/7)(x-0) which is simplified as y = (-15/7)x.
If the economy is currently facing a recession, what should the government do to minimize the effects of the recession?
In order to determine the best action the government can take to minimize the effects of the recession, the cause of the recession must be determined. If the recession is caused by a coordination failure, the government could introduce more money into the economy by increasing its spending to restore confidence. If the recession is caused by a supply shock, unless more supply is introduced, the government cannot cure the recession.