Join the conversation on IDNLearn.com and get the answers you seek from experts. Our platform provides prompt, accurate answers from experts ready to assist you with any question you may have.
Sagot :
Let's analyze the functions given for the released amounts of water from the two reservoirs.
Reservoir A:
[tex]\[ f(x) = x^2 - 7x + 5 \][/tex]
Reservoir B:
[tex]\[ g(x) = 3x^2 - 6x + 2 \][/tex]
The function representing the difference between the amounts of water released by the two reservoirs is:
[tex]\[ h(x) = f(x) - g(x) \][/tex]
First, let's find the expressions for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] when [tex]\( x = 1 \)[/tex] week:
For Reservoir A:
[tex]\[ f(1) = 1^2 - 7 \cdot 1 + 5 \][/tex]
[tex]\[ f(1) = 1 - 7 + 5 \][/tex]
[tex]\[ f(1) = -1 \][/tex]
For Reservoir B:
[tex]\[ g(1) = 3 \cdot 1^2 - 6 \cdot 1 + 2 \][/tex]
[tex]\[ g(1) = 3 - 6 + 2 \][/tex]
[tex]\[ g(1) = -1 \][/tex]
Now, let's calculate [tex]\( h(1) \)[/tex]:
[tex]\[ h(1) = f(1) - g(1) \][/tex]
[tex]\[ h(1) = -1 - (-1) \][/tex]
[tex]\[ h(1) = -1 + 1 \][/tex]
[tex]\[ h(1) = 0 \][/tex]
Given the results:
[tex]\[ f(1) = -1 \][/tex]
[tex]\[ g(1) = -1 \][/tex]
[tex]\[ h(1) = 0 \][/tex]
Let’s evaluate the given statements one-by-one:
1. [tex]\( h(x) = -2x^2 - 13x + 6 \)[/tex]:
Check if this is true for [tex]\( x = 1 \)[/tex]:
[tex]\[ h(1) = -2(1)^2 - 13(1) + 6 \][/tex]
[tex]\[ h(1) = -2 - 13 + 6 \][/tex]
[tex]\[ h(1) = -9 \neq 0 \][/tex]
The first statement is False.
2. [tex]\( h(x) = -2x^2 - x + 3 \)[/tex]:
Check if this is true for [tex]\( x = 1 \)[/tex]:
[tex]\[ h(1) = -2(1)^2 - 1(1) + 3 \][/tex]
[tex]\[ h(1) = -2 - 1 + 3 \][/tex]
[tex]\[ h(1) = 0 \][/tex]
The second statement is True.
3. Reservoir A releases less water than Reservoir B over 1 week.
Check if [tex]\( f(1) < g(1) \)[/tex]:
[tex]\[ f(1) = -1 \][/tex]
[tex]\[ g(1) = -1 \][/tex]
[tex]\[ -1 \not< -1 \][/tex]
This statement is False.
4. Reservoir A releases the same amount of water as Reservoir B over 1 week.
Check if [tex]\( f(1) = g(1) \)[/tex]:
[tex]\[ f(1) = -1 \][/tex]
[tex]\[ g(1) = -1 \][/tex]
[tex]\[ -1 = -1 \][/tex]
This statement is True.
5. Reservoir A releases more water than Reservoir B over 1 week.
Check if [tex]\( f(1) > g(1) \)[/tex]:
[tex]\[ f(1) = -1 \][/tex]
[tex]\[ g(1) = -1 \][/tex]
[tex]\[ -1 \not> -1 \][/tex]
This statement is False.
To summarize:
- [tex]\( h(x) = -2x^2 - 13x + 6 \)[/tex] is False.
- [tex]\( h(x) = -2x^2 - x + 3 \)[/tex] is True.
- Reservoir A releases less water than Reservoir B over 1 week is False.
- Reservoir A releases the same amount of water as Reservoir B over 1 week is True.
- Reservoir A releases more water than Reservoir B over 1 week is False.
Reservoir A:
[tex]\[ f(x) = x^2 - 7x + 5 \][/tex]
Reservoir B:
[tex]\[ g(x) = 3x^2 - 6x + 2 \][/tex]
The function representing the difference between the amounts of water released by the two reservoirs is:
[tex]\[ h(x) = f(x) - g(x) \][/tex]
First, let's find the expressions for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] when [tex]\( x = 1 \)[/tex] week:
For Reservoir A:
[tex]\[ f(1) = 1^2 - 7 \cdot 1 + 5 \][/tex]
[tex]\[ f(1) = 1 - 7 + 5 \][/tex]
[tex]\[ f(1) = -1 \][/tex]
For Reservoir B:
[tex]\[ g(1) = 3 \cdot 1^2 - 6 \cdot 1 + 2 \][/tex]
[tex]\[ g(1) = 3 - 6 + 2 \][/tex]
[tex]\[ g(1) = -1 \][/tex]
Now, let's calculate [tex]\( h(1) \)[/tex]:
[tex]\[ h(1) = f(1) - g(1) \][/tex]
[tex]\[ h(1) = -1 - (-1) \][/tex]
[tex]\[ h(1) = -1 + 1 \][/tex]
[tex]\[ h(1) = 0 \][/tex]
Given the results:
[tex]\[ f(1) = -1 \][/tex]
[tex]\[ g(1) = -1 \][/tex]
[tex]\[ h(1) = 0 \][/tex]
Let’s evaluate the given statements one-by-one:
1. [tex]\( h(x) = -2x^2 - 13x + 6 \)[/tex]:
Check if this is true for [tex]\( x = 1 \)[/tex]:
[tex]\[ h(1) = -2(1)^2 - 13(1) + 6 \][/tex]
[tex]\[ h(1) = -2 - 13 + 6 \][/tex]
[tex]\[ h(1) = -9 \neq 0 \][/tex]
The first statement is False.
2. [tex]\( h(x) = -2x^2 - x + 3 \)[/tex]:
Check if this is true for [tex]\( x = 1 \)[/tex]:
[tex]\[ h(1) = -2(1)^2 - 1(1) + 3 \][/tex]
[tex]\[ h(1) = -2 - 1 + 3 \][/tex]
[tex]\[ h(1) = 0 \][/tex]
The second statement is True.
3. Reservoir A releases less water than Reservoir B over 1 week.
Check if [tex]\( f(1) < g(1) \)[/tex]:
[tex]\[ f(1) = -1 \][/tex]
[tex]\[ g(1) = -1 \][/tex]
[tex]\[ -1 \not< -1 \][/tex]
This statement is False.
4. Reservoir A releases the same amount of water as Reservoir B over 1 week.
Check if [tex]\( f(1) = g(1) \)[/tex]:
[tex]\[ f(1) = -1 \][/tex]
[tex]\[ g(1) = -1 \][/tex]
[tex]\[ -1 = -1 \][/tex]
This statement is True.
5. Reservoir A releases more water than Reservoir B over 1 week.
Check if [tex]\( f(1) > g(1) \)[/tex]:
[tex]\[ f(1) = -1 \][/tex]
[tex]\[ g(1) = -1 \][/tex]
[tex]\[ -1 \not> -1 \][/tex]
This statement is False.
To summarize:
- [tex]\( h(x) = -2x^2 - 13x + 6 \)[/tex] is False.
- [tex]\( h(x) = -2x^2 - x + 3 \)[/tex] is True.
- Reservoir A releases less water than Reservoir B over 1 week is False.
- Reservoir A releases the same amount of water as Reservoir B over 1 week is True.
- Reservoir A releases more water than Reservoir B over 1 week is False.
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. IDNLearn.com is your go-to source for accurate answers. Thanks for stopping by, and come back for more helpful information.
