Example Question - conversion
Here are examples of questions we've helped users solve.
Complex Number Conversion to Polar Form
To convert the complex number \(\frac{1+7i}{(2-i)^2}\) to polar form, we first simplify the expression and then find its magnitude and angle. <p>Let's simplify \(\frac{1+7i}{(2-i)^2}\):</p> <p>\(= \frac{1+7i}{(2-i)(2-i)}\)</p> <p>\(= \frac{1+7i}{4 - 2i - 2i + i^2}\)</p> <p>\(= \frac{1+7i}{4 - 4i - 1}\) (since \(i^2 = -1\))</p> <p>\(= \frac{1+7i}{3 - 4i}\)</p> <p>Multiply the numerator and the denominator by the conjugate of the denominator:</p> <p>\(= \frac{(1+7i)(3+4i)}{(3-4i)(3+4i)}\)</p> <p>\(= \frac{3 + 4i + 21i + 28i^2}{9 + 12i - 12i - 16i^2}\)</p> <p>\(= \frac{3 + 25i - 28}{9 + 16}\) (since \(i^2 = -1\))</p> <p>\(= \frac{-25 + 25i}{25}\)</p> <p>\(= -1 + i\)</p> <p>The magnitude \(r\) is given by \(r = \sqrt{(-1)^2 + (1)^2}\):</p> <p>\(= \sqrt{1 + 1}\)</p> <p>\(= \sqrt{2}\)</p> <p>The angle \(\theta\) can be found from \(\tan(\theta) = \frac{1}{-1}\):</p> <p>\(\theta = \arctan(-1)\)</p> <p>We notice that the complex number lies in the second quadrant, hence \(\theta = \pi + \arctan(-1)\)</p> <p>\(\theta = \pi - \frac{\pi}{4}\)</p> <p>\(\theta = \frac{3\pi}{4}\)</p> <p>The polar form of the complex number is \(r(\cos(\theta) + i\sin(\theta))\):</p> <p>\(= \sqrt{2} \left(\cos\left(\frac{3\pi}{4}\right) + i\sin\left(\frac{3\pi}{4}\right)\right)\)</p>
Fractional Price Conversion
<p>\frac{3}{12} RM = \frac{3 \div 3}{12 \div 3} RM = \frac{1}{4} RM</p> <p>\frac{8}{48} RM = \frac{8 \div 8}{48 \div 8} RM = \frac{1}{6} RM</p>
Conversion and Division of Metric Units
<p>For part (a), we first convert 2.1 kg to grams since there are 1000 grams in a kilogram.</p> <p>2.1 kg = 2.1 \times 1000 g = 2100 g</p> <p>Now we divide 330 g by 2100 g.</p> <p>\frac{330 g}{2100 g} = \frac{330}{2100} = \frac{1}{2100/330} = \frac{1}{6.36363636} \approx 0.157</p> <p>The approximate answer for part (a) is 0.157.</p>
Calculating Height of Post Using Similar Triangles
Para resolver esta pregunta, podemos usar las proporciones dictadas por los triángulos semejantes formados por el poste, su sombra y los rayos de luz que llegan al suelo, y la varilla, su sombra y los rayos de luz que llegan al suelo. Dado que tenemos la sombra y la altura de la varilla, podemos establecer la misma relación con el poste y su sombra. La relación es la siguiente: Altura de la varilla / Longitud de la sombra de la varilla = Altura del poste / Longitud de la sombra del poste Sustituyendo los valores que tenemos obtenemos: 49 cm / 63 cm = Altura del poste / 4.5 m Primero, debemos convertir la sombra del poste de metros a centímetros para que las unidades sean consistentes: 4.5 m = 450 cm Ahora, podemos resolver para la altura del poste: 49 cm / 63 cm = Altura del poste / 450 cm (49 cm * 450 cm) / 63 cm = Altura del poste Calculamos el valor numérico: (49 * 450) / 63 = Altura del poste 22050 / 63 = Altura del poste 350 = Altura del poste Por lo tanto, la altura del poste es de 350 cm. Para expresar este valor en metros, dividimos por 100 ya que hay 100 cm en un metro: 350 cm / 100 = 3.5 m Por lo tanto, la altura del poste es de 3,5 metros. La respuesta correcta es la opción a) 3,5 m.
Pounds to Tons Conversion
The image shows a conversion task from pounds (lb) to tons (T). To convert pounds to tons, you need to know that one ton is equal to 2,000 pounds. To find out how many tons are in 5,500 pounds, you divide the number of pounds by the number of pounds in a ton. \[ 5,500 \text{ lb} \div 2,000 \text{ lb/ton} = 2.75 \text{ tons} \] So, 5,500 pounds is equal to 2.75 tons.
Conversion Between Exponential and Logarithmic Forms
The image shows a written question that asks to convert an equation from exponential form to logarithmic form and vice versa. The equation given is: 7^4 = 2401 Let's first convert this exponential form to logarithmic form. The general relationship between exponential and logarithmic forms is: If b^x = y, then log_b(y) = x Applying this to the equation 7^4 = 2401, it converts to: log_7(2401) = 4 This means the logarithm (base 7) of 2401 is equal to 4. Now let's convert from logarithmic form back to exponential form. Using the same relationship backward, the logarithmic form log_7(2401) = 4 converts to: 7^4 = 2401 So we have successfully demonstrated converting the equation from exponential to logarithmic form and back to exponential form.
Millimeters to Centimeters Conversion
Certainly! To convert millimeters to centimeters, you need to know that 10 millimeters make up 1 centimeter. So, to convert 45 millimeters to centimeters, you divide 45 by 10. \( 45 \text{ mm} \div 10 = 4.5 \text{ cm} \) Therefore, 45 millimeters equals 4.5 centimeters.
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