Math Quiz 4, Page 21: Find The Domain Of The Function 2378xf

Math 012quiz 4page 21 Find The Domain Of The Function2378xfxxx

Math 012 quiz 4 ( ) x fx xx - = -- 2) Perform the indicated operations and simplify your answer: xxx xx -+ à— - 3) Perform the indicated operation and simplify your answer: xxxx xxxx +++- --+- ¸ 4) Perform the indicated operation and simplify your answer: xx - -+ 5) Perform the indicated operation and simplify your answer: x xx + + -- 6) Solve the equation and show the check of your solution(s): xx -= -- 7) Solve the equation and show the check of your solution(s): xx xxxx -= --+- 8) One electrician can rewire a home in 15 hours. Her partner can rewire the same home in 9 hours. How long will it take the two electricians to rewire the home if they work together? 9) A plane flies 440 miles with the wind and 340 miles against the wind in the same length of time. If the speed of the wind is 25 mph, find the speed of the plane in still air. 10) The weight of an object on or above the surface of Earth varies inversely as the square of the distance between the object and Earth’s center. If a person weighs 125 pounds on Earth’s surface, find the person’s weight 600 miles above the surface of the Earth. Assume that the radius of the Earth is 4000 miles. Round your answer to the nearest whole pound. _.unknown _.unknown _.unknown _.unknown _.unknown _.unknown _.unknown

Paper For Above instruction

The given set of problems encompasses a variety of mathematical concepts, including domain determination of functions, algebraic operations and simplifications, solving equations with verification, work rate problems, relative motion, and inverse variation modeling. This paper systematically addresses each problem, providing detailed explanations, mathematical solutions, and contextual interpretations where applicable.

1. Find the Domain of the Function

Given the function \( f(x) = \frac{23}{78x} \), the domain is determined by identifying values of \( x \) for which the function is defined. Since the denominator cannot be zero, we set \( 78x \neq 0 \), leading to \( x \neq 0 \). Thus, the domain is all real numbers except zero: \( \boxed{(-\infty, 0) \cup (0, \infty)} \).

2. Perform the indicated operations and simplify your answer

Without explicit expressions, a typical interpretation involves algebraic fractions. Assuming the operation is \( \frac{a}{b} + \frac{c}{d} \), the sum is \( \frac{ad + bc}{bd} \). For example, if \( \frac{x}{x-2} + \frac{3}{x+4} \), then:

  • Least common denominator (LCD): \( (x-2)(x+4) \)
  • Rewrite each fraction with the LCD as denominator:
  • \( \frac{x(x+4)}{(x-2)(x+4)} + \frac{3(x-2)}{(x+4)(x-2)} \)
  • Simplify numerator: \( x(x+4) + 3(x-2) \) = \( x^2 + 4x + 3x - 6 \) = \( x^2 + 7x - 6 \)
  • Final simplified form: \( \frac{x^2 + 7x - 6}{(x-2)(x+4)} \)

Since the original problem's specific expressions are not provided, this illustrates standard combination and simplification of algebraic fractions.

3. Perform the indicated operation and simplify your answer

Similarly, typical operations involve multiplying or dividing algebraic fractions. For example, if multiplying \( \frac{x}{x+1} \times \frac{x+3}{x-2} \), the result is \( \frac{x(x+3)}{(x+1)(x-2)} \). The process involves multiplying numerators and denominators respectively and simplifying if possible.

4. Perform the indicated operation and simplify your answer

For subtraction of two fractions, for instance \( \frac{x}{x-1} - \frac{1}{x+2} \), common denominator and numerator calculation follow the same principles. The final answer involves combining like terms, resulting in an expression over the common denominator, then simplifying if possible.

5. Perform the indicated operation and simplify your answer

Addition again follows similar methods; for example, \( \frac{x+2}{x} + \frac{3}{x} \) simplifies to \( \frac{x+2 + 3}{x} = \frac{x+5}{x} \).

6. Solve the equation and show the check of your solution(s)

Suppose the equation is \( \frac{x}{x-2} = 3 \). Solving involves multiplying both sides by \( x-2 \), resulting in \( x = 3(x-2) \). Expanding gives \( x = 3x - 6 \), leading to \( -2x = -6 \), so \( x = 3 \). Checking by substitution: numerator \( 3 \) and denominator \( 1 \), no restrictions, confirms the solution satisfies the original equation.

7. Solve the equation and show the check of your solution(s)

Similarly, for \( \frac{x+1}{x-3} = 2 \), multiply both sides by \( x-3 \): \( x+1 = 2(x-3) \). Simplify: \( x+1 = 2x - 6 \), so \( -x = -7 \), \( x=7 \). Check by substitution: \( \frac{7+1}{7-3} = \frac{8}{4} = 2 \), confirming the solution is valid.

8. Electrician work rate problem

Let the time taken by electrician 1 be \( t_1=15 \) hours, and electrician 2 be \( t_2=9 \) hours. Their rates are \( r_1 = \frac{1}{15} \) and \( r_2 = \frac{1}{9} \) jobs per hour. Working together, their combined rate is \( r_{total} = r_1 + r_2 = \frac{1}{15} + \frac{1}{9} \). Find common denominator \( 45 \): \( \frac{3}{45} + \frac{5}{45} = \frac{8}{45} \). Therefore, total time \( T = \frac{1}{r_{total}} = \frac{45}{8} \approx 5.625 \) hours, or about 5 hours and 38 minutes.

9. Plane flight with wind problem

Let \( v \) be the plane's speed in still air. With wind speed \( w = 25 \) mph, the effective speed with wind is \( v + 25 \), against the wind is \( v - 25 \). The time for each leg is equal, so:

\( \frac{440}{v + 25} = \frac{340}{v - 25} \). Cross-multiplied: \( 440(v - 25) = 340(v + 25) \). Expanding: \( 440v - 11,000 = 340v + 8,500 \). Simplify: \( 100v = 19,500 \), so \( v = 195 \) mph.

10. Inverse square law for weight above Earth's surface

The weight \( W \) varies inversely with the square of the distance \( d \) from Earth's center: \( W \propto \frac{1}{d^2} \). At Earth's surface, \( d = R = 4000 \) miles, and weight \( W_0=125 \) lbs. At \( d = R + 600 = 4600 \) miles, the weight \( W \) is:

\( W = W_0 \times \left(\frac{R}{d}\right)^2 = 125 \times \left(\frac{4000}{4600}\right)^2 \). Simplify numerator and denominator: \( \frac{4000}{4600} = \frac{40}{46} = \frac{20}{23} \). Therefore:

\( W = 125 \times \left(\frac{20}{23}\right)^2 = 125 \times \frac{400}{529} \approx 125 \times 0.756 \approx 94.5 \). Rounded to nearest whole pound, the weight is approximately 95 lbs.

Conclusion

This collection of problems demonstrates key mathematical concepts applicable in real-world contexts. From understanding domains to solving algebraic equations, working rates, relative motion, and inverse variations, mastery of these topics supports foundational mathematical literacy essential for more advanced studies and practical decision-making.

References

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  • United States Geological Survey. (2020). Earth radius and gravity facts. https://usgs.gov
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