Respuesta :
Answer:
the correct answer is:
a. 77.5 meters
explanation:
Let's denote the width of the rectangle as \( w \) meters.
Given that the rectangle is three times as long as it is wide, the length (\( l \)) is \( 3w \) meters.
[tex]The formula for the area (\( A \)) of a rectangle is \( A = \text{length} \times \text{width} \).[/tex]
So, we have:
[tex]\[ A = l \times w \][/tex]
[tex]\[ 1800 = (3w) \times w \][/tex]
[tex]\[ 1800 = 3w^2 \][/tex]
Dividing both sides by 3:
[tex]\[ 600 = w^2 \][/tex]
Taking the square root of both sides:
[tex]\[ w = \sqrt{600} \][/tex]
[tex]\[ w \approx 24.5 \text{ meters} \][/tex]
So, the width of the rectangle is approximately 24.5 meters.
Now, the length of the rectangle is \( 3w \), so:
[tex]\[ l = 3 \times 24.5 = 73.5 \text{ meters} \][/tex]
The diagonals of a rectangle can be calculated using the formula:
[tex]\[ \text{Diagonal} = \sqrt{\text{length}^2 + \text{width}^2} \][/tex]
For our rectangle:
[tex]\[ \text{Diagonal} = \sqrt{73.5^2 + 24.5^2} \][/tex]
[tex]\[ \text{Diagonal} \approx \sqrt{5406.25 + 600.25} \][/tex]
[tex]\[ \text{Diagonal} \approx \sqrt{6006.5} \][/tex]
[tex]\[ \text{Diagonal} \approx 77.5 \text{ meters (rounded to the nearest tenth)} \][/tex]
So, the sum of the diagonals of the rectangle is approximately 77.5 meters.
Therefore, the correct answer is:
a. 77.5 meters
Final answer:
The rectangle's width is √600 meters, and length is 3×√600 meters. Using the Pythagorean theorem, the diagonal is 77.5 meters. The sum of both diagonals is 154.9 meters, rounded to the nearest tenth.So, the correct answer is option b.
Explanation:
To solve the problem, let’s denote the width of the rectangle as w meters, and the length as 3w meters (since it’s given that the rectangle is 3 times as long as it is wide). The area of the rectangle is given by the product of its length and width which is 1800 square meters.
The area A = length × width, therefore we can set up the equation:1800 = w × 3w = 3w². Dividing both sides by 3 gives w² = 600. Taking the square root of both sides gives w = √600, which is the width of the rectangle. Now, to find the length, we multiply the width by 3, so the length is 3√600. The length of the rectangle is therefore 3×√600 meters.To calculate the diagonals, we use the Pythagorean theorem, because a rectangle’s diagonals are equal, and the rectangle can be split into two right triangles. The length of the diagonal d can be found using:
d = √(length² + width²) = √((3w)² + (w)²) = √(9w² + w²) = √(10w²) = √(10·600) = √6000. After calculation, d is approximately 77.5 meters, which is the length of one diagonal. The sum of the diagonals is simply twice the length of one diagonal since both diagonals are equal. Thus, the sum of the diagonals is 2 × 77.5 = 154.9 meters.The correct answer is then b. 154.9 meters.
