Example Question - calculations
Here are examples of questions we've helped users solve.
Finding the Volume of a Rectangular Prism
<p>El volumen \( V \) de un prisma rectangular se calcula mediante la fórmula:</p> <p>\( V = largo \times ancho \times alto \)</p> <p>En este caso, todos los lados son \( 6 \, \text{cm} \):</p> <p>\( V = 6 \, \text{cm} \times 6 \, \text{cm} \times 6 \, \text{cm} \)</p> <p>\( V = 216 \, \text{cm}^3 \)</p>
Calculate the Sums
<p> Primero, encontramos un denominador común para las fracciones </p> <p> El denominador común de 4, 2 y 3 es 12. </p> <p> Convertimos las fracciones: </p> <p> \(\frac{3}{4} = \frac{9}{12}, \quad \frac{7}{2} = \frac{42}{12}, \quad \frac{7}{3} = \frac{28}{12}\) </p> <p> Luego, sumamos las fracciones: </p> <p> \(\frac{9}{12} + \frac{42}{12} + \frac{28}{12} = \frac{79}{12}\) </p> <p> La respuesta es \(\frac{79}{12}\). </p>
Multiplication Table Calculations for Various Values of z
Das Bild zeigt eine Tabelle, in der das doppelte, vierfache, zehnfache und achtfache der Zahl z berechnet werden soll, wobei z die Werte 12030, 1212 und 3333 annimmt. Zusätzlich gibt es zwei Multiplikationstabellen, die wahrscheinlich zur Hilfe dienen sollen. Lassen Sie uns diese Berechnungen durchführen: Für z = 12030: - Das Doppelte von z (2 * z) ist 2 * 12030 = 24060. - Das Vierfache von z (4 * z) ist 4 * 12030 = 48120. - Das Zehnfache von z (10 * z) ist 10 * 12030 = 120300. - Das Achtfache von z (8 * z) ist 8 * 12030 = 96240. Für z = 1212: - Das Doppelte von z (2 * z) ist 2 * 1212 = 2424. - Das Vierfache von z (4 * z) ist 4 * 1212 = 4848. - Das Zehnfache von z (10 * z) ist 10 * 1212 = 12120. - Das Achtfache von z (8 * z) ist 8 * 1212 = 9696. Für z = 3333: - Das Doppelte von z (2 * z) ist 2 * 3333 = 6666. - Das Vierfache von z (4 * z) ist 4 * 3333 = 13332. - Das Zehnfache von z (10 * z) ist 10 * 3333 = 33330. - Das Achtfache von z (8 * z) ist 8 * 3333 = 26664. Diese Ergebnisse können nun in die entsprechenden Felder der Tabelle eingetragen werden.
Finding Values for Continuity in a Piecewise Function
The image shows a mathematics question regarding a piecewise-defined function. The function \( f(x) \) is defined differently for different intervals of \( x \): \[ f(x) = \begin{cases} \log_3(3x + a) & , x < 1 \\ 3x & , x = 1 \\ \sqrt{x + b} & , x > 1 \end{cases} \] The question states that the function is continuous at \( x = 1 \) and asks for the values of \( a \) and \( b \). Continuity at \( x = 1 \) means that the left-hand limit as \( x \) approaches 1, the right-hand limit as \( x \) approaches 1, and the function's value at \( x = 1 \) must all be equal. Let's calculate these: 1. The left-hand limit as \( x \) approaches 1 is the limit of \( \log_3(3x + a) \) as \( x \) approaches 1 from the left: \[ \lim_{{x \to 1^-}} \log_3(3x + a) = \log_3(3 \cdot 1 + a) = \log_3(3 + a) \] 2. The function's value at \( x = 1 \) is given directly as \( f(1) = 3 \cdot 1 = 3 \). 3. The right-hand limit as \( x \) approaches 1 is the limit of \( \sqrt{x + b} \) as \( x \) approaches 1 from the right: \[ \lim_{{x \to 1^+}} \sqrt{x + b} = \sqrt{1 + b} \] For continuity at \( x = 1 \), we must have \( \log_3(3 + a) = 3 = \sqrt{1 + b} \). From here we can find the values of \( a \) and \( b \): \(\log_3(3 + a) = 3\) implies \(3^3 = 3 + a\), hence \( a = 27 - 3 = 24 \). \(\sqrt{1 + b} = 3\) implies \(1 + b = 3^2\), hence \( b = 9 - 1 = 8 \). Thus, \( a = 24 \) and \( b = 8 \).
Solving Area of Rectangles Inside Each Other
The image shows two rectangles, one inside the other. The larger rectangle has a length described as "x + 8" and a width described as "x + 5". The smaller rectangle, presumably positioned inside the larger one, has a length described as "x + 1" and a width described as "x". To solve a question involving these rectangles, we would typically be asked to express the area of one in terms of the other or find the dimensions of the rectangles given certain conditions. However, since the specific question isn't stated in the image, I'll provide a general approach. If we were to find the area of each rectangle, we would: 1. Calculate the area of the larger rectangle by multiplying its length and width: Area = (x + 8)(x + 5) 2. Calculate the area of the smaller rectangle by multiplying its length and width: Area = (x + 1)(x) Without additional information or a specific question, we cannot solve further. If you have a particular equation or condition that relates these rectangles, please provide that for further assistance.
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