0.8p = t At a store, a coat originally priced at p dollars is on sale for t dollars, and the relationship between p and t is given in the equation above. What is p in terms of t? a) p = t - 0.8 b) p = 0.8t c) p = 0.8/t d) p = t/0.8
This a rewriting equations problem. You have to rewrite the equation so that it is a p= equation instead of a t= equation. To save time, it is helpful to skip the description they normally provide with the problem and go straight to the algebra. You use the same skills for solving for p when there are other variables present as you would in a regular single-equation problem. In other words, treat t as though it were a number and solve for p by isolating it and using opposite operations. To isolate p, you have to get rid of the 0.8 that is in front of it. Because the 0.8 is being multiplied to p, the opposite operation you need to do is dividing the 0.8 on both sides. When you divide 0.8 on both sides, the 0.8's cancel out on the left side and you are left with p = t / 0.8 for the answer (d).
Find the intervals where the following function is increasing: f(x) = (1/3)x^3 + 2x^2 - 5x - 6.
The function is increasing when f'(x) > 0. So first, use the sum, difference, and power rules to get the derivative: f'(x) = x^2 + 4x - 5. Then find the critical points by setting the derivative equal to zero and solving for x. When you set x^2 + 4x - 5 equal to 0, you can factor the quadratic function into binomials: (x + 5)(x - 1) = 0. The critical points are therefore -5 and 1 and the endpoints are -infinity and infinity. The intervals we will look at are (-infinity, -5), (-5, 1), and (1, infinity). Take a number in each interval and plug it into the derivative function to see if it is increasing on that interval. For (-infinity, -5), we could choose x= -6. Then f'(x) = 7, which means the function is increasing on that interval because f'(x) > 0. For (-5, 1), we could choose x= 0. Then f'(x) = -5, which means the function is decreasing on that interval because f'(x) < 0. Finally, we could choose x= 2 for (1, infinity). Then f(x) = 7, which means the function is increasing on that interval because f'(x) > 0. It follows that the correct answer will be (-infinity, -5) U (1, infinity).
A price floor for a good will most likely cause: a) an increase in demand for the good b) a shortage of the good c) lower prices for the good d) a surplus of the good e) a decrease in the supply of the good
A price floor is when the government sets prices above the market equilibrium. Because of the higher price, suppliers will be willing to produce more.. This will increase the quantity supplied of the good. But the higher price will turn off buyers, which will decrease the quantity demanded of the good. Because quantity supplied will be greater than quantity supplied, there will be a surplus of the good (answer choice d).