A model of carry-on luggage has a length that is 5 inches greater than its depth. Airline regulations require the sum of length, width, and depth cannot exceed 57 inches. These con... A model of carry-on luggage has a length that is 5 inches greater than its depth. Airline regulations require the sum of length, width, and depth cannot exceed 57 inches. These conditions, with the assumption that this sum is 57 inches, can be modeled by the function V(x) that gives the luggage volume, in cubic inches, in terms of its depth, x, in inches. If its volume is 1500 cubic inches, determine two possibilities for its depth.
Understand the Problem
The question is asking to find the depth of a model of carry-on luggage based on its volume and the relationship between its dimensions. We need to set up an equation based on the provided conditions and solve for the depth, x.
Answer
The depth of the carry-on luggage is approximately \( 7.37 \, \text{cm} \).
Answer for screen readers
The depth of the carry-on luggage is approximately ( 7.37 , \text{cm} ).
Steps to Solve
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Identify the Known Values Given the volume of the luggage is ( V = 2400 , \text{cm}^3 ), and that the height ( h ) and width ( w ) have a specific relationship with the depth ( x ). The dimensions of the luggage are given as:
- Height ( h = 2x )
- Width ( w = 3x )
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Volume Formula The volume ( V ) of a rectangular prism (luggage in this case) can be calculated with the formula: $$ V = h \times w \times x $$ Substituting the expressions for ( h ) and ( w ): $$ V = (2x) \times (3x) \times x $$
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Set Up the Equation Substituting the known volume into the equation: $$ 2400 = (2x) \times (3x) \times x $$ This simplifies to: $$ 2400 = 6x^3 $$
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Solve for ( x^3 ) To isolate ( x^3 ), divide both sides by 6: $$ x^3 = \frac{2400}{6} $$ Calculating the right side gives: $$ x^3 = 400 $$
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Find the Value of ( x ) Now, take the cube root of both sides to find ( x ): $$ x = \sqrt[3]{400} $$
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Calculate the Result Using a calculator or estimating, we find: $$ x \approx 7.37 , \text{cm} $$
The depth of the carry-on luggage is approximately ( 7.37 , \text{cm} ).
More Information
The volume of a rectangular prism is crucial in applications like luggage dimensions, ensuring we can maximize space without exceeding limits. The relationship between dimensions helps create a mathematical model that represents real-world objects.
Tips
- Miscalculating the volume or confusing it with the surface area.
- Forgetting to cube the depth when applying the volume formula.
- Incorrect substitution of values into the volume equation.
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