Let's solve the problem step-by-step:
### Part (a)
Simplify the expression [tex]\((-13.2) + 8.1\)[/tex].
To simplify this expression, follow the basic rules of arithmetic addition:
[tex]\[
-13.2 + 8.1 = -5.1
\][/tex]
### Part (b)
How are [tex]\((-13.2) + 8.1\)[/tex] and [tex]\(13.2 + (-8.1)\)[/tex] related? Explain without computing.
The expressions [tex]\((-13.2) + 8.1\)[/tex] and [tex]\(13.2 + (-8.1)\)[/tex] are related by the commutative property of addition. The commutative property states that the order in which two numbers are added does not change the sum. In essence, [tex]\(a + b = b + a\)[/tex]. If we understand this property, we can see that:
[tex]\[
(-13.2) + 8.1 = 8.1 + (-13.2)
\][/tex]
and
[tex]\[
13.2 + (-8.1) = (-8.1) + 13.2
\][/tex]
Therefore, both expressions represent sums where the order of the numbers has been rearranged according to the commutative property.
### Part (c)
Using a property of operations, what can you say about the sum of the two expressions?
From Part (b), based on the commutative property of addition, we know that the sum of [tex]\((-13.2) + 8.1\)[/tex] and [tex]\(13.2 + (-8.1)\)[/tex] is the same. This is because the commutative property guarantees that the order of addition does not affect the result. So:
[tex]\[
(-13.2) + 8.1 = 13.2 + (-8.1)
\][/tex]
Both sums are equal, reinforcing the idea that the expressions are essentially the same in value.
### PART B
On which day did the temperature change more? Explain your reasoning.
To find out which day had a greater temperature change, we need to calculate the difference between the temperatures at sunset and sunrise for each day.
Day 1:
[tex]\[
\text{Temperature change at Day 1} = \text{Sunset} - \text{Sunrise} = 13.49 - (-11.31)
\][/tex]
When we subtract a negative number, it's equivalent to adding the absolute value of that number:
[tex]\[
13.49 - (-11.31) = 13.49 + 11.31 = 24.8
\][/tex]
Day 2:
[tex]\[
\text{Temperature change at Day 2} = \text{Sunset} - \text{Sunrise} = 25.25 - (-7.69)
\][/tex]
Similarly, subtracting a negative number results in addition:
[tex]\[
25.25 - (-7.69) = 25.25 + 7.69 = 32.94
\][/tex]
Comparing the two temperature changes:
[tex]\[
24.8 \text{ on Day 1} \quad \text{and} \quad 32.94 \text{ on Day 2}
\][/tex]
Therefore, the temperature changed more on Day 2.
### Conclusion
In conclusion:
- The simplified value of [tex]\((-13.2) + 8.1\)[/tex] is [tex]\(-5.1\)[/tex].
- The expressions [tex]\((-13.2) + 8.1\)[/tex] and [tex]\(13.2 + (-8.1)\)[/tex] are related by the commutative property of addition, making them equal in value.
- The temperature change was greater on Day 2 with a change of [tex]\(32.94^\circ \text{F}\)[/tex] compared to [tex]\(24.8^\circ \text{F}\)[/tex] on Day 1.
