Sprint 18081 and solutions

Dec 13, 2019

Click here for the Wednesday Dec 11 sprint 18081.

Problem 1:

The top three steps of a stairway are shown. The stairway has 61 steps with 61 risers that are each 4 cm high. The 78 treads are each 10 cm wide. If all the stairs were shown in the figure, what would be the area of the figure?

Problem 2:

The seats in a theater are set in 32 rows in such a way that each row after the first has 8 more seats than the previous row. If the first row has 25 seats, how many seats are in the theater?

The first row has 25 seats. Each row after the first seat has 8 more seats more than the row before it. So the second row has 25+8. The third row has 25+8+8 seats. The 4th row has 25+8+8+8 seats.
In general, the n'th row in the theater has 25+(n-1)*8 seats. .
That's an arithmetic sequence pattern. You are being asked for the sum of an arithmetic sequence.
The formula for the sum of an arithmetic sequence we use is SNFL or `\Sigma = n((F+L)/2) `

In this problem, the n is the number of rows in the theater, F and L are the number of seats in the first and last row. Setting up the variables:

n=32 The number or rows in the theater or terms in the arithmetic sequence.
F=25 The number of seats in the first row or the first term in the arithmetic sequence.
L is the last term or the number of seats in the 32nd row. Using the rule 25+(n+1)*8, the last row has 25+(32-1)*8 or 273 seats.
So L=273
Plug the values into the formula:
`\Sigma = 32((25+273)/2)`
`\Sigma = 4768` The number of seats in the theater is 4768.

Problem 3:

What is the single discount that is equivalent to the two successive discounts of 29% off followed by 28% off the discounted_price? Answer to the nearest hundredth of a percent.

Problem 4:

The seats in a theater are set in 28 rows in such a way that each row after the first has 4 more seats than the previous row. If the first row has 28 seats, how many seats are in the 26th row?