The problem describes a carpet consisting of two parts, blue and white. The carpet's dimensions are given as $x$ cm and 60 cm. The blue part is a right-angled triangle. The perimeter of the carpet is 2402 cm. The goal is to calculate the area, in cm$^2$, of the blue part of the carpet.

1. Problem Description

The problem describes a carpet consisting of two parts, blue and white. The carpet's dimensions are given as $x$ cm and 60 cm. The blue part is a right-angled triangle. The perimeter of the carpet is 2402 cm. The goal is to calculate the area, in cm$^2$, of the blue part of the carpet.

2. Solution Steps

First, calculate the length of the hypotenuse of the triangle (which is the blue part) using the Pythagorean theorem. Let the hypotenuse be $h$.

$h = \sqrt{x^2 + 60^2}$

Second, use the given perimeter to find $x$.

The perimeter is the sum of all sides: $x + 60 + x + h = 2402$.

Substitute the expression for $h$:

$2x + 60 + \sqrt{x^2 + 60^2} = 2402$

$\sqrt{x^2 + 60^2} = 2402 - 2x - 60$

$\sqrt{x^2 + 3600} = 2342 - 2x$

Square both sides:

$x^2 + 3600 = (2342 - 2x)^2$

$x^2 + 3600 = 2342^2 - 2 * 2342 * 2x + 4x^2$

$x^2 + 3600 = 5484964 - 9368x + 4x^2$

$0 = 3x^2 - 9368x + 5481364$

Divide the equation by 3:

$x^2 - \frac{9368}{3}x + \frac{5481364}{3} = 0$

$x^2 - 3122.67x + 1827121.33 = 0$

Since $2402 \approx 2x + 60 + \sqrt{x^2}$, where the root is roughly equal to $x$,

then $2402 \approx 3x + 60$,

so $3x \approx 2342$,

so $x \approx \frac{2342}{3} \approx 780.67$

Let's use the perimeter equation to find $x$:

$2x + 60 + \sqrt{x^2 + 3600} = 2402$

$\sqrt{x^2 + 3600} = 2342 - 2x$

Let $x=781$:

$\sqrt{781^2 + 3600} = \sqrt{609961 + 3600} = \sqrt{613561} = 783.3$

$2342 - 2(781) = 2342 - 1562 = 780$

The value of $x$ is approximately

7

8

1. Let's assume that $x$ is close to

7

8

1.

Third, compute the area of the blue part (right-angled triangle):

$Area = \frac{1}{2} * base * height = \frac{1}{2} * x * 60 = 30x$

If $x = 781$, then $Area = 30 * 781 = 23430$ cm$^2$.

3. Final Answer

23430 cm$^2$