Example Question - calculating areas
Here are examples of questions we've helped users solve.
Calculating Areas of Pyramid's Base and Lateral Faces
The image displays a math problem that involves finding the areas of different parts of a pyramid. It already contains a calculation for the area of the base, as well as the calculation and answer for the area of one of the lateral faces. Looking at the image, we can see that the base is a square because all sides are of equal length (4.5 in). The area of a square is found using the formula A = s^2, where s is the length of a side of the square. The calculation in the image is correct: A = (4.5 in)^2 = 20.25 in^2. That is the area of the base of the pyramid. For a lateral face, which is a triangle in this case, the area is calculated using the formula A = 1/2 * b * h, where b is the base of the triangle and h is the height. From the image, it shows they've used one side of the square base as the base of the triangle (b = 4.5 in) and the slant height of the pyramid as the height of the triangle (h = 7 in). The calculation for one lateral face is A = 1/2 * (4.5 in) * (7 in) = 15.75 in^2. This is also correct as shown in the image. To sum it up: - The area of the base (A_base) is 20.25 in^2. - The area of one lateral face (A_lateral) is 15.75 in^2. Since a pyramid with a square base has four identical lateral faces, if you wanted to find the total area of all the lateral faces, you would multiply the area of one lateral face by four (4 * 15.75 in^2 = 63 in^2). The image does not show this total calculation, but I included it here for a more comprehensive understanding.
Calculating Areas of Base and Lateral Faces of Right Triangular Pyramid
The image shows a right triangular pyramid with a square base. The sides of the square base are 4.5 inches, and the slant height (the height of each triangular face, not the height from the base to the apex) is 4.4 inches. Part A of the question asks for the area of the base and the area of each lateral face. To find the area of the base (A_base), we use the formula for the area of a square: \[ A_{\text{base}} = s^2 \] where \( s \) is the length of a side of the square. Since the side length is 4.5 inches, we calculate the area as: \[ A_{\text{base}} = (4.5\ \text{in})^2 = 20.25\ \text{in}^2 \] To find the area of each lateral face (A_face), we use the formula for the area of a triangle: \[ A_{\text{face}} = \frac{1}{2} \times \text{base} \times \text{height} \] For the lateral faces, the base is the side of the square (4.5 inches), and the height is the slant height (4.4 inches). The area of each lateral face is: \[ A_{\text{face}} = \frac{1}{2} \times 4.5\ \text{in} \times 4.4\ \text{in} = 9.9\ \text{in}^2 \] So, the area of the base is 20.25 square inches, and the area of each lateral face is 9.9 square inches.
Calculating Areas of Colored Shapes
Claro, vamos a calcular el área de la parte coloreada de las figuras que se muestran en la imagen. 1. Para la primera figura, que es un círculo dividido en partes iguales, con una parte coloreada: - El área total del círculo es π * r^2. Aquí, r (el radio) es de 3 cm (la mitad de 6 cm). - Entonces, el área total es π * (3 cm)^2 = π * 9 cm^2. - Como el círculo está dividido en 8 partes iguales y una de esas partes está coloreada, dividimos el área total entre 8. - Área de la parte coloreada = (π * 9 cm^2) / 8 = (9/8)π cm^2. 2. Para la segunda figura, que es un sector de un círculo (una forma de "pastel" o "queso"): - Primero calculamos el área total del círculo como π * r^2, donde r es 10 cm. - El área total del círculo es π * (10 cm)^2 = π * 100 cm^2. - Observando la figura, parece que el sector coloreado es un cuarto del círculo (90 grados de 360 grados), así que dividimos el área total entre 4. - Área de la parte coloreada = (π * 100 cm^2) / 4 = 25π cm^2. Entonces, las áreas coloreadas son (9/8)π cm^2 para la primera figura y 25π cm^2 para la segunda figura.
Calculating Volumes and Areas
Let's solve both parts of this question one by one: ### Part a) To calculate the volume of the pyramid with a square base: The formula for the volume of a pyramid is \( \frac{1}{3} \times \text{base area} \times \text{height} \). Given: - The height (\(h\)) of the pyramid is 10 cm. - The side (\(s\)) of the square base is 5 cm. First, calculate the base area: - Base area = \( s^2 = 5 \text{ cm} \times 5 \text{ cm} = 25 \text{ cm}^2 \). Next, calculate the volume of the pyramid: - Volume = \( \frac{1}{3} \times \text{base area} \times h = \frac{1}{3} \times 25 \text{ cm}^2 \times 10 \text{ cm} \). - Volume = \( \frac{1}{3} \times 25 \times 10 \text{ cm}^3 = \frac{250}{3} \text{ cm}^3 \approx 83.33 \text{ cm}^3 \). To convert cubic centimeters to liters, remember that 1 liter equals 1000 cubic centimeters: - Volume in liters = \( \frac{83.33}{1000} \text{ liters} \approx 0.083 \text{ liters} \). So the volume of the pyramid is approximately 0.083 liters (to 2 decimal places: 0.08 liters). ### Part b) For the rectangular cover plate with rounded corners to form sectors: Given: - The dimensions of the rectangular plate are 500 mm by 300 mm. - Each corner is rounded to form a sector with a radius of 25 mm. Firstly, calculate the area of the original rectangle: - Area of rectangle = Length × Width = 500 mm × 300 mm = 150000 mm². Next, you have to subtract the area of the four sectors. Each sector is a quarter of a circle with a radius of 25 mm. The area of one full circle with radius \(r\) is \( \pi r^2 \). Therefore, the area of one quarter-circle (or sector) would be \( \frac{1}{4} \pi r^2 \). - Area of one sector = \( \frac{1}{4} \pi (25 \text{ mm})^2 \). - Area of one sector = \( \frac{1}{4} \pi \times 625 \text{ mm}^2 \). - Area of one sector = \( 156.25 \pi \text{ mm}^2 \). Since there are four such sectors: - Total area of the four sectors = \( 4 \times 156.25 \pi \text{ mm}^2 \). - Total area of the four sectors = \( 625 \pi \text{ mm}^2 \). Subtracting this from the area of the rectangle gives the final area: - Final area of the plate = Area of rectangle - Total area of the four sectors. - Final area of the plate = 150000 mm² - 625π mm². - Final area of the plate = \( 150000 \text{ mm}^2 - 1963.4954 \text{ mm}^2 \) (using \( \pi \approx 3.1416 \)). - Final area of the plate ≈ 147036.5 mm² (rounded to one decimal place). So, the final area of the rectangular cover plate, after rounding off the corners to form sectors, is approximately 147036.5 mm².
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