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Sagot :
Certainly! Let's solve the problem step-by-step.
### Given:
- The perimeter (P) of the rectangle is 60 cm.
- The area (A) of the rectangle is 216 cm².
- Let the length of the rectangle be [tex]\( x \)[/tex] and the breadth be [tex]\( y \)[/tex].
### a) What is [tex]\( x + y \)[/tex]?
#### Step 1: Using the Perimeter Formula
The perimeter of a rectangle is given by:
[tex]\[ P = 2(x + y) \][/tex]
Given [tex]\( P = 60 \)[/tex]:
[tex]\[ 60 = 2(x + y) \][/tex]
#### Step 2: Solving for [tex]\( x + y \)[/tex]
To find [tex]\( x + y \)[/tex]:
[tex]\[ x + y = \frac{60}{2} \][/tex]
[tex]\[ x + y = 30 \][/tex]
So, [tex]\( x + y \)[/tex] is 30 cm.
### b) What is [tex]\( x \times x \times y \)[/tex]?
To find [tex]\( x \times x \times y \)[/tex], we need to determine the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] first.
#### Step 3: Using the Area Formula
The area of a rectangle is given by:
[tex]\[ A = x \times y \][/tex]
Given [tex]\( A = 216 \)[/tex]:
[tex]\[ x \times y = 216 \][/tex]
#### Step 4: Solving for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]
We now have two equations:
1. [tex]\( x + y = 30 \)[/tex]
2. [tex]\( x \times y = 216 \)[/tex]
We can use these to form a quadratic equation. From the first equation:
[tex]\[ y = 30 - x \][/tex]
Substitute [tex]\( y \)[/tex] into the second equation:
[tex]\[ x \times (30 - x) = 216 \][/tex]
[tex]\[ 30x - x^2 = 216 \][/tex]
[tex]\[ x^2 - 30x + 216 = 0 \][/tex]
This is a quadratic equation which can be solved for [tex]\( x \)[/tex].
#### Step 5: Solving the Quadratic Equation
The quadratic equation:
[tex]\[ x^2 - 30x + 216 = 0 \][/tex]
We solve it using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -30 \)[/tex], and [tex]\( c = 216 \)[/tex].
Discriminant [tex]\( \Delta = b^2 - 4ac \)[/tex]:
[tex]\[ \Delta = (-30)^2 - 4 \cdot 1 \cdot 216 \][/tex]
[tex]\[ \Delta = 900 - 864 \][/tex]
[tex]\[ \Delta = 36 \][/tex]
So,
[tex]\[ x = \frac{30 \pm \sqrt{36}}{2} \][/tex]
[tex]\[ x = \frac{30 \pm 6}{2} \][/tex]
Thus,
[tex]\[ x = \frac{30 + 6}{2} = 18 \][/tex]
[tex]\[ x = \frac{30 - 6}{2} = 12 \][/tex]
If [tex]\( x = 18 \)[/tex], then [tex]\( y = 30 - 18 = 12 \)[/tex].
If [tex]\( x = 12 \)[/tex], then [tex]\( y = 30 - 12 = 18 \)[/tex].
Both pairs [tex]\((18, 12)\)[/tex] and [tex]\((12, 18)\)[/tex] satisfy the equations.
#### Step 6: Calculating [tex]\( x \times x \times y \)[/tex]
We now calculate [tex]\( x \times x \times y \)[/tex]:
For [tex]\( x = 18 \)[/tex] and [tex]\( y = 12 \)[/tex]:
[tex]\[ x \times x \times y = 18 \times 18 \times 12 = 3888 \][/tex]
For [tex]\( x = 12 \)[/tex] and [tex]\( y = 18 \)[/tex]:
[tex]\[ x \times x \times y = 12 \times 12 \times 18 = 3888 \][/tex]
So, [tex]\( x \times x \times y \)[/tex] is 3888 cm³.
### Summary:
a) [tex]\( x + y = 30 \)[/tex] cm
b) [tex]\( x \times x \times y = 3888 \)[/tex] cm³
### Given:
- The perimeter (P) of the rectangle is 60 cm.
- The area (A) of the rectangle is 216 cm².
- Let the length of the rectangle be [tex]\( x \)[/tex] and the breadth be [tex]\( y \)[/tex].
### a) What is [tex]\( x + y \)[/tex]?
#### Step 1: Using the Perimeter Formula
The perimeter of a rectangle is given by:
[tex]\[ P = 2(x + y) \][/tex]
Given [tex]\( P = 60 \)[/tex]:
[tex]\[ 60 = 2(x + y) \][/tex]
#### Step 2: Solving for [tex]\( x + y \)[/tex]
To find [tex]\( x + y \)[/tex]:
[tex]\[ x + y = \frac{60}{2} \][/tex]
[tex]\[ x + y = 30 \][/tex]
So, [tex]\( x + y \)[/tex] is 30 cm.
### b) What is [tex]\( x \times x \times y \)[/tex]?
To find [tex]\( x \times x \times y \)[/tex], we need to determine the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] first.
#### Step 3: Using the Area Formula
The area of a rectangle is given by:
[tex]\[ A = x \times y \][/tex]
Given [tex]\( A = 216 \)[/tex]:
[tex]\[ x \times y = 216 \][/tex]
#### Step 4: Solving for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]
We now have two equations:
1. [tex]\( x + y = 30 \)[/tex]
2. [tex]\( x \times y = 216 \)[/tex]
We can use these to form a quadratic equation. From the first equation:
[tex]\[ y = 30 - x \][/tex]
Substitute [tex]\( y \)[/tex] into the second equation:
[tex]\[ x \times (30 - x) = 216 \][/tex]
[tex]\[ 30x - x^2 = 216 \][/tex]
[tex]\[ x^2 - 30x + 216 = 0 \][/tex]
This is a quadratic equation which can be solved for [tex]\( x \)[/tex].
#### Step 5: Solving the Quadratic Equation
The quadratic equation:
[tex]\[ x^2 - 30x + 216 = 0 \][/tex]
We solve it using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -30 \)[/tex], and [tex]\( c = 216 \)[/tex].
Discriminant [tex]\( \Delta = b^2 - 4ac \)[/tex]:
[tex]\[ \Delta = (-30)^2 - 4 \cdot 1 \cdot 216 \][/tex]
[tex]\[ \Delta = 900 - 864 \][/tex]
[tex]\[ \Delta = 36 \][/tex]
So,
[tex]\[ x = \frac{30 \pm \sqrt{36}}{2} \][/tex]
[tex]\[ x = \frac{30 \pm 6}{2} \][/tex]
Thus,
[tex]\[ x = \frac{30 + 6}{2} = 18 \][/tex]
[tex]\[ x = \frac{30 - 6}{2} = 12 \][/tex]
If [tex]\( x = 18 \)[/tex], then [tex]\( y = 30 - 18 = 12 \)[/tex].
If [tex]\( x = 12 \)[/tex], then [tex]\( y = 30 - 12 = 18 \)[/tex].
Both pairs [tex]\((18, 12)\)[/tex] and [tex]\((12, 18)\)[/tex] satisfy the equations.
#### Step 6: Calculating [tex]\( x \times x \times y \)[/tex]
We now calculate [tex]\( x \times x \times y \)[/tex]:
For [tex]\( x = 18 \)[/tex] and [tex]\( y = 12 \)[/tex]:
[tex]\[ x \times x \times y = 18 \times 18 \times 12 = 3888 \][/tex]
For [tex]\( x = 12 \)[/tex] and [tex]\( y = 18 \)[/tex]:
[tex]\[ x \times x \times y = 12 \times 12 \times 18 = 3888 \][/tex]
So, [tex]\( x \times x \times y \)[/tex] is 3888 cm³.
### Summary:
a) [tex]\( x + y = 30 \)[/tex] cm
b) [tex]\( x \times x \times y = 3888 \)[/tex] cm³
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