Example Question - shaded area
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Calculate the Area of a Shaded Region in a Combined Shape
The shaded area is made up of a rectangular part and a quarter of a circular sector (since we're given a right angle and the arc of a circle with center P). The rectangle has one side that is 7 m and shares the other side with the radius of the sector, which is 7 m. The area of the rectangle \( A_{rectangle} \) is: \[ A_{rectangle} = \text{length} \times \text{width} \] \[ A_{rectangle} = 7 \times 7 \] \[ A_{rectangle} = 49 \ m^2 \] The area of the circular sector \( A_{sector} \) is a quarter of the area of a circle with radius 7 m: \[ A_{sector} = \frac{1}{4} \pi r^2 \] \[ A_{sector} = \frac{1}{4} \pi (7)^2 \] \[ A_{sector} = \frac{1}{4} \pi (49) \] \[ A_{sector} = 12.25 \pi \ m^2 \] Therefore, the total shaded area \( A \) is the sum of the rectangle area and the sector area: \[ A = A_{rectangle} + A_{sector} \] \[ A = 49 + 12.25 \pi \] \[ A \approx 49 + 38.48 \] \[ A \approx 87.48 \ m^2 \] So, the area of the shaded region is approximately 87.48 square meters.
Calculating Shaded Areas in Composite Figures
<p>Para la figura (d):</p> <p>Area del cuadrado = l^2</p> <p>Area del círculo = \pi r^2</p> <p>El lado del cuadrado (l) es igual al diámetro del círculo, entonces r = \frac{l}{2}</p> <p>Área sombreada = Área del cuadrado - Área del círculo</p> <p>Área sombreada = l^2 - \pi \left(\frac{l}{2}\right)^2</p> <p>Área sombreada = l^2 - \frac{\pi l^2}{4}</p> <p>Área sombreada = \left(1 - \frac{\pi}{4}\right)l^2</p> <p>Como l = 14 metros, sustituimos y calculamos:</p> <p>Área sombreada = \left(1 - \frac{\pi}{4}\right)(14)^2</p> <p>Área sombreada = (1 - \frac{3.1416}{4}) \cdot 196</p> <p>Área sombreada = (1 - 0.7854) \cdot 196</p> <p>Área sombreada = 0.2146 \cdot 196</p> <p>Área sombreada = 42.0624 \text{ metros cuadrados}</p> <p>La respuesta es aproximadamente 42.0624 metros cuadrados.</p>
Area Calculation of Overlapping Squares
Let the side of the larger square be \( x \) cm. The area of the larger square is \( x^2 \) cm\(^2\). Given the side of the smaller square is 10 cm, the area of the smaller square is \( 10^2 = 100 \) cm\(^2\). The difference in area between the shaded part B and the unshaded part A is given as 24 cm\(^2\). We know that the unshaded part A is a smaller square with an area of 100 cm\(^2\) minus the part overlapping with the larger square. Let the side of the overlapped square be \( y \) cm. The area of the shaded part B includes two rectangles and the overlapped square: \( 2 \times x \times y + y^2 \). The difference given is: \( 2xy + y^2 - 100 = 24 \). As the side of the larger square is \( x \) and the side of the overlapped square is \( y \), then \( x = 10 + y \). Replace \( x \) with \( 10 + y \) in the difference equation: \( 2(10 + y)y + y^2 - 100 = 24 \) \( 20y + 2y^2 + y^2 - 100 = 24 \) \( 3y^2 + 20y - 124 = 0 \) Solving this quadratic equation by the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 3 \), \( b = 20 \), and \( c = -124 \): \( y = \frac{-20 \pm \sqrt{400 + 1488}}{6} \) \( y = \frac{-20 \pm \sqrt{1888}}{6} \) \( y = \frac{-20 \pm 2\sqrt{472}}{6} \) \( y = \frac{-10 \pm \sqrt{472}}{3} \) Since \( y \) must be positive and it is the smaller dimension, we take the positive root: \( y = \frac{-10 + \sqrt{472}}{3} \) Now we find \( x \): \( x = 10 + y \) \( x = 10 + \frac{-10 + \sqrt{472}}{3} \) The area of the unshaded part A is the side of the smaller square squared: Area of A is \( 10^2 = 100 \) cm\(^2\). The area of the shaded part B is the side of the larger square squared minus the area of A. Using the values of \( x \) and \( y \) found above, we can calculate the exact area of B.
Overlapping Square Areas Problem
<p>Let the side of the larger square be \( x \) cm, therefore its area is \( x^2 \) cm².</p> <p>Let the side of the smaller square be \( (x - 10) \) cm, area is \( (x - 10)^2 \) cm².</p> <p>The area of the shaded part A is the difference between the areas of the two squares: \( x^2 - (x - 10)^2 = 24 \) cm².</p> <p>Solving the equation:</p> <p>\( x^2 - (x^2 - 20x + 100) = 24 \)</p> <p>\( x^2 - x^2 + 20x - 100 = 24 \)</p> <p>\( 20x - 100 = 24 \)</p> <p>\( 20x = 124 \)</p> <p>\( x = 6.2 \) cm</p> <p>The side of the smaller square is \( (6.2 - 10) = -3.8 \) cm, which is not possible.</p> <p>There seems to be an error because the side of the smaller square cannot be a negative number. Please verify the information provided in the question.</p>
Integral Calculation for Shaded Area Between Linear Functions
La imagen muestra un gráfico con dos funciones lineales y un área sombreada entre ellas. Para encontrar el área del sector sombreado, necesitas integrar la diferencia de las dos funciones en el intervalo donde se superponen. Las funciones parecen ser f(x) = x y la otra es una recta positiva, posiblemente g(x) = kx, donde k es la pendiente. Sin valores específicos para los límites del área sombreada o la pendiente k, no puedo proporcionar una respuesta numérica exacta. Sin embargo, puedo explicarte cómo establecer la integral. Para el área sombreada, necesitas establecer A = ∫(g(x) - f(x)) dx La integral se calcula desde el punto de intersección de las dos rectas hasta el valor de x donde deseas detener el área. El punto de intersección de las rectas se da cuando f(x) = g(x), o sea, x = kx, lo cual sucede solo si x = 0 (asumiendo que k no es 1). Por lo tanto, el límite inferior de la integral es 0. Suponiendo que el límite superior es h, la integral es A = ∫[0,h](kx - x) dx A = ∫[0,h](x(k - 1)) dx Integrando x en el intervalo de 0 a h obtenemos A = [(k - 1)/2] x^2 | desde 0 hasta h A = [(k - 1)/2] h^2 Para darte una respuesta precisa, necesitaría los valores exactos de k (la pendiente de la segunda recta) y h (el límite superior del área sombreada).
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