Answer :
Certainly! Let's go through the problem step-by-step to understand the solution.
### Step 1: Define Variables
Let \( x \) represent the length of the shortest side of the flower-bed.
### Step 2: Express the Other Sides in Terms of \( x \)
- One side of the triangle is twice the length of the shortest side. Therefore, the length of this side is \( 2x \).
- The third side is 15 feet longer than the shortest side. Therefore, the length of this side is \( x + 15 \).
### Step 3: Set Up the Perimeter Equation
The perimeter of a triangle is the sum of the lengths of all three sides. According to the problem, the perimeter is 100 feet. Therefore, we can write the equation as:
[tex]\[ x + 2x + (x + 15) = 100 \][/tex]
### Step 4: Simplify the Equation
Combine like terms to simplify the equation:
[tex]\[ x + 2x + x + 15 = 100 \][/tex]
[tex]\[ 4x + 15 = 100 \][/tex]
### Step 5: Solve for \( x \)
Isolate \( x \) by performing the following steps:
1. Subtract 15 from both sides:
[tex]\[ 4x + 15 - 15 = 100 - 15 \][/tex]
[tex]\[ 4x = 85 \][/tex]
2. Divide both sides by 4:
[tex]\[ x = \frac{85}{4} \][/tex]
[tex]\[ x = 21.25 \][/tex]
So, the length of the shortest side is \( 21.25 \) feet.
### Step 6: Find the Lengths of the Other Sides
Now, we can determine the lengths of the other sides:
- The second side is \( 2x \):
[tex]\[ 2 \cdot 21.25 = 42.5 \][/tex] feet
- The third side is \( x + 15 \):
[tex]\[ 21.25 + 15 = 36.25 \][/tex] feet
### Conclusion
The lengths of the three sides of the flower-bed triangle are:
- Shortest side: \( 21.25 \) feet
- Second side: \( 42.5 \) feet
- Third side: \( 36.25 \) feet
### Answering the Equation Question
The correct equation that we used to set up the problem is:
[tex]\[ x + 2x + (x + 15) = 100 \][/tex]
Thus, the correct answer is:
[tex]\[ A) \, x + 2x + x + 15 = 100 \][/tex]
### Step 1: Define Variables
Let \( x \) represent the length of the shortest side of the flower-bed.
### Step 2: Express the Other Sides in Terms of \( x \)
- One side of the triangle is twice the length of the shortest side. Therefore, the length of this side is \( 2x \).
- The third side is 15 feet longer than the shortest side. Therefore, the length of this side is \( x + 15 \).
### Step 3: Set Up the Perimeter Equation
The perimeter of a triangle is the sum of the lengths of all three sides. According to the problem, the perimeter is 100 feet. Therefore, we can write the equation as:
[tex]\[ x + 2x + (x + 15) = 100 \][/tex]
### Step 4: Simplify the Equation
Combine like terms to simplify the equation:
[tex]\[ x + 2x + x + 15 = 100 \][/tex]
[tex]\[ 4x + 15 = 100 \][/tex]
### Step 5: Solve for \( x \)
Isolate \( x \) by performing the following steps:
1. Subtract 15 from both sides:
[tex]\[ 4x + 15 - 15 = 100 - 15 \][/tex]
[tex]\[ 4x = 85 \][/tex]
2. Divide both sides by 4:
[tex]\[ x = \frac{85}{4} \][/tex]
[tex]\[ x = 21.25 \][/tex]
So, the length of the shortest side is \( 21.25 \) feet.
### Step 6: Find the Lengths of the Other Sides
Now, we can determine the lengths of the other sides:
- The second side is \( 2x \):
[tex]\[ 2 \cdot 21.25 = 42.5 \][/tex] feet
- The third side is \( x + 15 \):
[tex]\[ 21.25 + 15 = 36.25 \][/tex] feet
### Conclusion
The lengths of the three sides of the flower-bed triangle are:
- Shortest side: \( 21.25 \) feet
- Second side: \( 42.5 \) feet
- Third side: \( 36.25 \) feet
### Answering the Equation Question
The correct equation that we used to set up the problem is:
[tex]\[ x + 2x + (x + 15) = 100 \][/tex]
Thus, the correct answer is:
[tex]\[ A) \, x + 2x + x + 15 = 100 \][/tex]
