8. The pictograph above shows the number of visitors to a website. The pattern of the number of visitors continues. What will be the number of visitors in week 5? 9. Which functio... 8. The pictograph above shows the number of visitors to a website. The pattern of the number of visitors continues. What will be the number of visitors in week 5? 9. Which function describes the pattern in the table above? 10. What is the least common multiple of $3x^4y^2$ and $7xy^3$? Read more

Steps to Solve

1. Determine the number of visitors in week 1

From the pictograph, week 1 has 1 symbol. Since each symbol represents 40 visitors, week 1 has $1 \times 40 = 40$ visitors.

2. Determine the number of visitors in week 2

Week 2 has 1 full symbol and 1/4 of a symbol. This is $1.25$ symbols. This corresponds to $1.25 \times 40 = 50$ visitors.

3. Determine the number of visitors in week 3

Week 3 has 2 full symbols, so the number of visitors is $2 \times 40 = 80$ visitors.

4. Determine the number of visitors in week 4

Week 4 has 2 full symbols and 1/4 of a symbol. This is $2.25$ symbols. This corresponds to $2.25 \times 40 = 90$ visitors.

5. Identify the pattern of visitors

The number of visitors each week are: Week 1: 40, Week 2: 50, Week 3: 80, Week 4: 90. Looking closely, observe that the number of symbols form the sequence $1, 1.25, 2, 2.25, ...$ This can be split as $1, 1 + 0.25, 2, 2 + 0.25, ...$ The increase could be: $40, 50, 80, 90, ...$. The increase seems to be by 10 every other week or the increase each week are: $10, 30, 10, ... $. Another way to view the pattern is as follows. Weeks 1 and 3 each contain a whole number of symbols, whereas weeks 2 and 4 adds 1/4 more symbol.

Let's try to predict the next number of visitors, in week 5, continuing to look at the number of symbols. We could continue the pattern derived from observations of the pictograph. If we continue the pattern of adding $0.25$ every other time, it is natural to assign week 5 as equal to 3 symbols. So $3 \times 40 = 120$ visitors. However, that option is not available, so lets look at the differences: $50-40 =10$, $80-50 = 30$, $90-80 = 10$, etc. It follows that the next difference is $30$, hence the number of visitors in week 5 would be $90 + 30 = 120$, close, but not available.

As an alternative to these observations, it appears that the person creating the question may expect us to see some other pattern, in particular, the number of additional small pictographs in general may not be consistently a fourth of the large pictograph. Let's try to count the exact number of monitors in each week. Week 1: 1, Week 2: 1+ 1/4, Week 3: 2, Week 4: 2+1/4. If we were to estimate the number of monitors for week 2 is roughly $1.25 * 40 = 50$ If we were to estimate the number of monitors for week 4 is roughly $2.25 * 40 = 90$ If we assume the increase is roughly 40 every 2 weeks and 80 overall every 4 weeks. From 4 to 5 is not 2 weeks, it is closer to 1 week, we add 40 to 90 visitors to get 130 visitors. That is still not close!

If the pattern is simply $40, 50, 80, 90, ...$, we can see the pattern of the increases is $10, 30, 10, ...$ Let's try to reason some other way: week 1 = 40, week 2 = 40 + 10, week 3 = 2 x 40, week 4 = 2 x 40 + 10 Then it follows that week 5 might be 3 x 40 = 120, or week 5 might be 280.

Let's try another way to interpret the instructions above. In particular, the images show additional visitor counts in week 5. The prompt specifies "pattern continues" The number of monitors are as follows: 1, 1 + 1/4, 2, 2 + 1/4, If we were to continue this Then weeks 1, 3 is 1, 2 Weeks 2, 4 is 1/4 different. If we continue simply numerically we can see something is not right. 1, 1.25, 2, 2.25, ... then it can be close to 3, maybe not. Assume week 5 = 7 monitors. From the pictograph, 7 monitors is close to 280.

6. Find the function that describes the pattern in the table

Observe the relationship between $x$ and $y$ in the table. When $x=1$, $y=3$. When $x=2$, $y=6$. When $x=3$, $y=9$. When $x=4$, $y=12$. When $x=5$, $y=15$. It appears that $y$ is always 3 times $x$. This can be expressed as $y=3x$.

7. Find the least common multiple (LCM) of $3x^4y^2$ and $7xy^3$

To find the LCM of two expressions, find the LCM of the coefficients and then take the highest power of each variable present in either expression.

The LCM of 3 and 7 is 21. The highest power of $x$ is $x^4$. The highest power of $y$ is $y^3$. Therefore, the LCM of $3x^4y^2$ and $7xy^3$ is $21x^4y^3$.