Homework 4

• Binomial coefficients and binomial theorem
  • Find the first three terms (starting with x¹⁰) in the binomial expansion of (x-2y)¹⁰.
  • Use the first few terms of the binomial expansion of (1+0.002)¹⁰⁰ to estimate the value of 1.002¹⁰⁰ correct to two decimal places.
    Hint: start with just one term and try using more and more terms. See how the approximation grows and decide when to stop.
• Rates, proportional reasoning
  • 5 cows can eat 2 acres of grass in 10 days. How many days will it take 10 cows to eat 6 acres of grass?
  • Pooh, Pigglet, Tigger, and Roo ate a cake. They took turns, eating it just one person at a time. Each of them was eating exactly for the period of time that it would take the other three to eat half of the cake. How much faster would they be done eating this cake if they were eating all at once instead of taking turns (assuming each of them would eat at the same pace)?
    Clarifications:
    (1) They ate the whole cake.
    (2) They could switch more than three times, but the total amount of time that each of them was eating was equal to the amount of time that it would take the other three to eat half of the cake. So by rearranging their turns if necessary, we can assume that each of them ate their portion of the cake in one sitting. E.g. first Pooh ate for the time that it would take Pigglet, Tigger, and Roo to eat half of the cake. Then Pigglet ate for the time that it would take Pooh, Tigger, and Roo to eat half of the cake. Then it was Tigger's turn. Then Roo's turn. It happened so that exactly at the end of Roo's turn, the whole cake was gone.
    (3) The problem does not say that all four rates are equal. Only that each person always eats at the same rate, no matter whether they are alone at the table or with their friends.