Example Question - rectangle area
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Geometry Problem: Calculating the Length of a Rectangle's Diagonal
<p>El enunciado proporciona la información de que la base del rectángulo mide 3 metros, su perímetro es de 18 metros, y nos pide calcular la longitud de la diagonal del rectángulo.</p> <p>Primero, calcularemos la longitud del otro lado del rectángulo utilizando el perímetro (P).</p> <p>Sea \( l \) la longitud del lado desconocido del rectángulo, entonces:</p> <p>\[ P = 2 \cdot base + 2 \cdot l \]</p> <p>\[ 18 = 2 \cdot 3 + 2 \cdot l \]</p> <p>\[ 18 = 6 + 2l \]</p> <p>\[ 2l = 18 - 6 \]</p> <p>\[ 2l = 12 \]</p> <p>\[ l = 6 \text{ metros} \]</p> <p>Ahora que conocemos ambos lados del rectángulo, podemos calcular la medida de la diagonal (d) utilizando el teorema de Pitágoras:</p> <p>\[ d^2 = base^2 + l^2 \]</p> <p>\[ d^2 = 3^2 + 6^2 \]</p> <p>\[ d^2 = 9 + 36 \]</p> <p>\[ d^2 = 45 \]</p> <p>\[ d = \sqrt{45} \]</p> <p>\[ d = \sqrt{9 \cdot 5} \]</p> <p>\[ d = 3\sqrt{5} \text{ metros} \]</p> <p>Por lo tanto, la longitud de la diagonal del rectángulo es \( 3\sqrt{5} \) metros.</p>
Calculating Area of a Rose Garden
To find the area of this rose garden, you need to calculate the area of the rectangle and add the areas of the two semi-circles on either end of the rectangle. First, calculate the area of the rectangle. The area \( A_{rectangle} \) is given by the formula: \[ A_{rectangle} = \text{length} \times \text{width} \] For this garden: \[ A_{rectangle} = 21 \text{ ft} \times 14 \text{ ft} = 294 \text{ ft}^2 \] Now calculate the area of a full circle and then take half of it to get the area of a semi-circle. The diameter of the circles is the width of the rectangle, which is 14 feet. The radius \( r \) is therefore 7 feet (since radius is half the diameter). The area \( A_{circle} \) of a full circle is given by the formula: \[ A_{circle} = \pi r^2 \] But since we have two semi-circles, their combined area will equal the area of one full circle. So we will use the formula for a full circle and not divide it by 2 in the end. Let's calculate this area using \( \pi \approx 3.14 \): \[ A_{circle} = 3.14 \times 7^2 = 3.14 \times 49 = 153.86 \text{ ft}^2 \] So, to find the total area of the garden \( A_{garden} \), add the area of the rectangle and the area of the circles: \[ A_{garden} = A_{rectangle} + A_{circle} \] \[ A_{garden} = 294 \text{ ft}^2 + 153.86 \text{ ft}^2 = 447.86 \text{ ft}^2 \] Now, round to the nearest whole number if necessary and do not round your answer: \[ A_{garden} \approx 448 \text{ ft}^2 \] Therefore, the area of the garden is approximately 448 square feet.
Calculating Area of Polygon
To find the area of the polygon, you need to treat it as a combination of simpler shapes, such as rectangles or squares. Based on the figure provided, it looks like a rectangle with a smaller rectangle on top of it. First, find the area of the larger rectangle. The large rectangle has a length of 6 units and a height of 7.2 units. Area of large rectangle = length × height = 6 × 7.2 = 43.2 square units Next, find the area of the smaller rectangle on the top. It has a length of 3 units and a height of 1 unit. Area of small rectangle = length × height = 3 × 1 = 3 square units Now, add the area of the larger rectangle to the area of the smaller rectangle to get the total area of the polygon. Total area of the polygon = Area of large rectangle + Area of small rectangle Total area of the polygon = 43.2 + 3 = 46.2 square units The area of the polygon is 46.2 square units.
Solving for Area and Perimeter of a Rectangle with Variables
The image you've provided shows a rectangle with its sides labeled in terms of x and y: The longer side (length) is labeled as "4x - y" and "x + 6y". The shorter side (width) is labeled as "3x + 1". To calculate the area of the rectangle, we multiply the length by the width. However, we first need to make sure both expressions for the length are equal. Let's set them equal to each other and solve for one of the variables: 4x - y = x + 6y 4x - x = 6y + y 3x = 7y x = 7y / 3 Now we have a relationship between x and y. To find the area, we could substitute x into one of the sides and then calculate the area. But since we do not have specific values for x and y, we cannot calculate a numeric value for the area without more information. For the perimeter, you add up all the sides: Perimeter (P) = 2(length) + 2(width) P = 2(4x - y) + 2(3x + 1) P = 8x - 2y + 6x + 2 P = 14x - 2y + 2 You can then replace x with (7y / 3) in the perimeter equation to express the entire perimeter in terms of y: P = 14(7y / 3) - 2y + 2 To simplify this: P = (98/3)y - 2y + 2 P = (98y - 6y) / 3 + 2 P = (92y / 3) + 2 Thus, we've simplified the perimeter equation, but without specific numeric values for x and y, we can't find an exact numerical value for the perimeter. It's worth noting that there might be a bit of confusion because the rectangle is labeled with two expressions for its length that seem to be equal without having additional information about the relationship between x and y besides the equation we've derived. Normally, you would need either a specific value or an additional equation to solve for x and y and hence calculate the area and perimeter with numeric values.
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