Answer :
Sure, let's analyze each situation and determine whether they illustrate a quadratic equation. I'll provide a step-by-step solution for each.
### Situation 1:
Description:
The length of a swimming pool is 8 meters longer than its width, and the area of the pool is 105 square meters.
1. Define Variables:
Let the width of the swimming pool be [tex]\( w \)[/tex] meters.
2. Determine the Length:
According to the description, the length will be [tex]\( w + 8 \)[/tex] meters.
3. Calculate the Area:
The area [tex]\( A \)[/tex] of a rectangle (which is the shape of the pool) is given by the formula
[tex]\[ A = \text{length} \times \text{width} \][/tex]
Plug in the values:
[tex]\[ 105 = w \times (w + 8) \][/tex]
4. Form the Equation:
[tex]\[ w \times (w + 8) = 105 \][/tex]
Expand and rearrange the equation to standard form [tex]\( ax^2 + bx + c = 0 \)[/tex]:
[tex]\[ w^2 + 8w - 105 = 0 \][/tex]
Conclusion:
This equation is a quadratic equation because it is in the form [tex]\( w^2 + 8w - 105 = 0 \)[/tex], which fits the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex].
### Situation 2:
Description:
A garden measuring 7 meters by 12 meters will be expanded with a border of flowers. The border will be the same width [tex]\( x \)[/tex] meters around the entire garden and the border area will be 92 square meters.
1. Define Variables:
Let the width of the border be [tex]\( x \)[/tex] meters.
2. Determine New Dimensions:
After adding the border, the total length and total width of the garden will be:
- Total length = [tex]\( 7 + 2x \)[/tex]
- Total width = [tex]\( 12 + 2x \)[/tex]
3. Calculate the Original Area:
The original area of the garden is:
[tex]\[ \text{Original Area} = 7 \times 12 = 84 \text{ square meters} \][/tex]
4. Calculate the Expanded Area:
The expanded area of the garden including the border is the original area plus the border area:
[tex]\[ 84 + 92 = 176 \text{ square meters} \][/tex]
5. Form the Equation:
The total area after expansion is given by:
[tex]\[ (\text{Total length}) \times (\text{Total width}) = 176 \][/tex]
Substitute the expressions for total length and width:
[tex]\[ (7 + 2x) \times (12 + 2x) = 176 \][/tex]
Expand and rearrange the equation to standard form:
[tex]\[ 4x^2 + 14x + 24x + 84 = 176 \][/tex]
Combine like terms:
[tex]\[ 4x^2 + 38x + 84 = 176 \][/tex]
Subtract 176 from both sides to get:
[tex]\[ 4x^2 + 38x - 92 = 0 \][/tex]
Conclusion:
This equation is a quadratic equation because it is in the form [tex]\( 4x^2 + 38x - 92 = 0 \)[/tex], which fits the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex].
Both situations indeed describe quadratic equations, and we have justified this by representing each situation with a mathematical sentence that takes the form of a quadratic equation.
### Situation 1:
Description:
The length of a swimming pool is 8 meters longer than its width, and the area of the pool is 105 square meters.
1. Define Variables:
Let the width of the swimming pool be [tex]\( w \)[/tex] meters.
2. Determine the Length:
According to the description, the length will be [tex]\( w + 8 \)[/tex] meters.
3. Calculate the Area:
The area [tex]\( A \)[/tex] of a rectangle (which is the shape of the pool) is given by the formula
[tex]\[ A = \text{length} \times \text{width} \][/tex]
Plug in the values:
[tex]\[ 105 = w \times (w + 8) \][/tex]
4. Form the Equation:
[tex]\[ w \times (w + 8) = 105 \][/tex]
Expand and rearrange the equation to standard form [tex]\( ax^2 + bx + c = 0 \)[/tex]:
[tex]\[ w^2 + 8w - 105 = 0 \][/tex]
Conclusion:
This equation is a quadratic equation because it is in the form [tex]\( w^2 + 8w - 105 = 0 \)[/tex], which fits the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex].
### Situation 2:
Description:
A garden measuring 7 meters by 12 meters will be expanded with a border of flowers. The border will be the same width [tex]\( x \)[/tex] meters around the entire garden and the border area will be 92 square meters.
1. Define Variables:
Let the width of the border be [tex]\( x \)[/tex] meters.
2. Determine New Dimensions:
After adding the border, the total length and total width of the garden will be:
- Total length = [tex]\( 7 + 2x \)[/tex]
- Total width = [tex]\( 12 + 2x \)[/tex]
3. Calculate the Original Area:
The original area of the garden is:
[tex]\[ \text{Original Area} = 7 \times 12 = 84 \text{ square meters} \][/tex]
4. Calculate the Expanded Area:
The expanded area of the garden including the border is the original area plus the border area:
[tex]\[ 84 + 92 = 176 \text{ square meters} \][/tex]
5. Form the Equation:
The total area after expansion is given by:
[tex]\[ (\text{Total length}) \times (\text{Total width}) = 176 \][/tex]
Substitute the expressions for total length and width:
[tex]\[ (7 + 2x) \times (12 + 2x) = 176 \][/tex]
Expand and rearrange the equation to standard form:
[tex]\[ 4x^2 + 14x + 24x + 84 = 176 \][/tex]
Combine like terms:
[tex]\[ 4x^2 + 38x + 84 = 176 \][/tex]
Subtract 176 from both sides to get:
[tex]\[ 4x^2 + 38x - 92 = 0 \][/tex]
Conclusion:
This equation is a quadratic equation because it is in the form [tex]\( 4x^2 + 38x - 92 = 0 \)[/tex], which fits the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex].
Both situations indeed describe quadratic equations, and we have justified this by representing each situation with a mathematical sentence that takes the form of a quadratic equation.
