A ball is dropped from a height of 18m, and each rebound is 1/3 of the previous distance. a. how high is the ball in its 4th rebound? b. what is the total distance the ball has travelled before coming to rest?
2.Suppose that Joe has saved P50 the first week of the month, P150 on the second week, P450 on the third week, and so on.
a. How much will he save on the 15th week?
b. What will be his total savings for 8 weeks?
3.A particular substance decays in such a way that it loses one-fourth of its weight each day. If the substance originally weighs 500 grams, how much is left after 10 days?
4. The sides of a square are each 16cm long. A second square is inscribed by joining the midpoints of the sides, successively. In the second square, we repeat the process inscribing a third square. If this process is continued indefinitely, what is the sum of the areas of all the squares?
(GEOMETRIC SEQUENCE)
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Answer:
a. To find the height of the ball in its 4th rebound, we can use the given information that each rebound is 1/3 of the previous distance. Starting with a height of 18m, the distances for each rebound would be:
Therefore, the ball would be at a height of approximately 0.22 meters in its 4th rebound.
b. To calculate the total distance the ball has traveled before coming to rest, we need to sum up the distances traveled during each rebound. Since the distances form a geometric sequence with a common ratio of 1/3, we can use the formula for the sum of a geometric series:
Using the given information that the first term is 18m and the common ratio is 1/3, we can calculate the sum:
Therefore, the total distance traveled by the ball before coming to rest is 27 meters.
2.
a. To find how much Joe will save on the 15th week, we can observe that the amount he saves follows a pattern: P50, P150, P450, and so on. Each week, the savings amount triples. We can use this pattern to determine the savings for the 15th week:
...
Week 15: P450 * 3^14
Therefore, Joe will save P450 multiplied by 3 to the power of 14 on the 15th week.
b. To calculate Joe's total savings for 8 weeks, we can sum up the savings for each week. Using the same pattern, we have:
Therefore, Joe's total savings for 8 weeks would be P109,350.
3. If a substance loses one-fourth of its weight each day, we can calculate how much is left after 10 days by multiplying the original weight by (3/4) raised to the power of 10:
4. To find the sum of the areas of all the squares, we can use the concept of a geometric series. The first square has sides of 16cm, and subsequent squares are formed by joining the midpoints of the sides. This process continues indefinitely, creating a sequence of squares with side lengths forming a geometric progression.
The sum of an infinite geometric series can be calculated using the formula:
Where 'a' is the first term and 'r' is the common ratio.
Using the formula, we can calculate the sum of the areas of all the squares.