The equation A(w)=w^2+5w represents the relationship between the area in square units and the width of a rectangle whose length is 5 units longer than its width. Select the sentence that describes an accurate relationship between A and w .
1. A(w) has a minimum value at w=5 .
2. A(w) has a maximum value at w=5 .
3. A(w) increases as w increases for 0<w<5 .
4. A(w) increases as w increases when w>0 .
Option 1. A(w) has a minimum value at w=5.
The correct sentence that describes an accurate relationship between A and w is:
4. A(w) increases as w increases when w>0.
To determine the relationship between A(w) and w, we need to analyze the equation A(w) = w^2 + 5w.
First, we can note that the equation represents the area of a rectangle, where the length is 5 units longer than the width.
Now, let's evaluate the options:
1. A(w) has a minimum value at w=5.
To find the minimum value of A(w), we can take the derivative of the equation with respect to w and set it equal to zero. However, in this case, we are not given any information that suggests the minimum value occurs at w=5. Therefore, this option is incorrect.
2. A(w) has a maximum value at w=5.
Similar to option 1, we don't have any evidence that suggests a maximum value occurs at w=5. Hence, this option is also incorrect.
3. A(w) increases as w increases for 0 < w < 5.
To determine if this option is valid, we can analyze the equation. By substituting values within the given range, let's say for w=1 and w=4, we can calculate A(1) and A(4). If A(1) is smaller than A(4), then this option holds true.
A(1) = 1^2 + 5(1) = 1 + 5 = 6
A(4) = 4^2 + 5(4) = 16 + 20 = 36
Since A(1) = 6 and A(4) = 36, we can see that A(w) indeed increases as w increases for 0 < w < 5. Therefore, this option is correct.
4. A(w) increases as w increases when w > 0.
This statement is a generalization of the previous option (option 3) as it covers a wider range of values for w. Since option 3 was already verified, option 4 is also correct.
In conclusion, the correct options are 3. A(w) increases as w increases for 0 < w < 5 and 4. A(w) increases as w increases when w > 0.