The masses of 10 books are found to be in an arithmetic sequence. If their total mass is 13 kg and the lightest book has a mass of 400 grams, what is the mass of the heaviest book?
2,000 grams
2,100 grams
2,200 grams
2,300 grams
2,700 grams
Sn = n/2 (a+a+(n-1)d)
If Tn is the nth term,
S10 = 5(a+T10)
13000 = 5(400+T10)
T10 = 2200
The heaviest book is 2200g
Step 1: Convert the mass of the lightest book from grams to kilograms.
The lightest book has a mass of 400 grams. To convert grams to kilograms, divide by 1000:
400 grams ÷ 1000 = 0.4 kg.
Step 2: Determine the common difference of the arithmetic sequence.
Since the masses of the books are in an arithmetic sequence, we can determine the common difference by dividing the total mass of all the books by the number of books.
Total mass = 13 kg
Number of books = 10
Common difference = Total mass / Number of books
Common difference = 13 kg / 10 = 1.3 kg.
Step 3: Calculate the mass of the heaviest book.
The mass of the heaviest book can be determined by adding the common difference to the mass of the lightest book.
Mass of the heaviest book = Mass of lightest book + (Number of books - 1) × Common difference
Mass of the heaviest book = 0.4 kg + (10 - 1) × 1.3 kg
Mass of the heaviest book = 0.4 kg + 9 × 1.3 kg
Mass of the heaviest book = 0.4 kg + 11.7 kg
Mass of the heaviest book = 12.1 kg.
Step 4: Convert the mass of the heaviest book from kilograms to grams.
To convert the mass from kilograms to grams, multiply by 1000:
Mass of the heaviest book = 12.1 kg × 1000 = 12,100 grams.
Therefore, the mass of the heaviest book is 12,100 grams.
To find the mass of the heaviest book, we need to determine the common difference of the arithmetic sequence.
Given that the lightest book has a mass of 400 grams, we can convert it to kilograms by dividing by 1000: 400 grams / 1000 = 0.4 kg.
Next, we need to find the total mass of all the books, which is given as 13 kg. Since the lightest book has a mass of 0.4 kg, the sum of the masses of the remaining 9 books would be 13 kg - 0.4 kg = 12.6 kg.
To find the common difference, we can subtract the first term (0.4 kg) from the last term (mass of the heaviest book) and divide it by the number of terms minus one (9 books):
Common difference = (mass of the heaviest book - 0.4 kg) / (9 books)
Since the common difference is the difference between consecutive terms, we multiply it by 9 and add it to the mass of the lightest book to find the mass of the heaviest book:
Mass of the heaviest book = 0.4 kg + 9 × common difference
Let's calculate the common difference using the given total mass of the books.
12.6 kg = 9 × common difference
common difference = 12.6 kg / 9 = 1.4 kg
Finally, we can calculate the mass of the heaviest book:
Mass of the heaviest book = 0.4 kg + 9 × 1.4 kg = 0.4 kg + 12.6 kg = 13 kg
The mass of the heaviest book is 13 kg, which, when converted to grams, is equal to 13 kg × 1000 = 13,000 grams.
Therefore, the correct answer is 13,000 grams.