a. Jimmy can wash his car in 30 minutes. His 4-year old son, Jimmy Jr., takes 2
ID: 2919595 • Letter: A
Question
a. Jimmy can wash his car in 30 minutes. His 4-year old son, Jimmy Jr., takes 2 hours to wash the car. Working together, how long will it take them to wash the car? b. Find the dimensions of a right triangle where the legs and the hypotenuse are consecutive even integers. Hint: use the pythagorean theorem c. It takes Jimmy 45 minutes to file the monthly invoices. If Jimmy Jr. takes twice as long as Jimmy, how long will it take them working together to file invoices? d. One leg of a right triangle is 2 inches shorter than the longer leg. The hypotenuse is 2 inches longer than the longer leg. Find the length of each side. f. The sum of the squares of two consecutive even numbers is 100. Find the numbers.Explanation / Answer
a. Jimmy can wash the car in 30 minutes so that he can wash 1/30th of the car in 1 minute. His 4 year old son can wash the car in 120 minutes so that he can wash 1/120th of the car in 1 minute.Together, they can wash 1/30 +1/120 = 5/120 = 1/24th of the car in 1 minute. Hence, Jimmy and his son, together, can wash the car in 24 minutes.
b. Let the legs of the right triangle be a and a+2 and let its hypotenuse be a+4, where a is a non-zero positive even number. Now, as per the Pythagoras theorem, the square of the hypotenuse = sum of the squares of the other 2 legs so that (a+4)2 = a2 +(a+2)2 or, a2+8a+16 = a2+a2+4a+4 or, a2-4a-12 = 0 or, a2-6a+2a-12 = 0 or, a(a-6)+2(a-6) = 0 or, (a-6)(a+2) = 0. Now, since acannot be negative, hence a = 6. Thus the dimensions of the lega and the hypotenuse are 6,8 and 12 respectively.
c. Jimmy can file the monthly invoices in 45 minutes, so he can file 1/45 of the monthly invoices in 1 minute. Jimmy Jr. can file the monthly invoices in 90 minutes, so he can file 1/90 of the monthly invoices in 1 minute.Thus, Jimmy and Jimmy Jr., together can file 1/45 +1/90 = 3/90 = 1/30 of the monthly invoices in 1 minute. Hence, together, they can file the monthly invoices in 30 minutes
d. Let the longer leg of the right triangle be a. Then the shorter leg and the hypotenuse are a-2 and a+2 respectively. Now, as per the Pythagoras theorem, the square of the hypotenuse = sum of the squares of the other 2 legs so that (a+2)2 = a2 +(a-2)2 or, a2+4a+4 = a2+a2-4a+4 or, a2-8a = 0 or, a(a-8)=0 or a = 8 as a cannot be zer0. Thus, the sides of the right triangle are 6,8 and 10 respectively.
f.. Let the even numbers be a and a+2. Then a2+(a+2)2 = 100 or, 2a2+4a+4= 100 or, 2a2+4a-96 = 0 or, a2+2a-48 = 0 or, a2+8a-6a-48 = 0 or, a(a+8)-6(a+8) = 0 or, (a+8)(a-6) = 0. Thus, a = -8 or, a = 6. Thus, the required numbers are either -8 and -6 or, 6 and 8.
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