Joanne bought 3 baseball cards for $160. if the first card was as expensive as the combined price of the second and the third cards, and the second card cost $20 more than the third card, what is the price of the first card?

Let the price of the 3rd card be c. Now we can write the equation:
c + (c + 20) + (c + c + 20) = 160 (1)
Simplifying the left hand side of equation (1), we get:
4c + 40 = 160 (2)
Subtracting 40 from both sides of equation (2) gives:
4c = 120
from which c = $30.
Therefore the price of the second card = 30 + 20 = $50
and the price of the first card = 30 + 50 = $80.

Sin(a)+cos(a)+cot(a)-1=2(csc(x)+1)

Answer

1 number

explanation

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.