the volume of a solid can be expressed as ( v = 7x^{3} ). the volume is to be calculated with an error of no…

the volume of a solid can be expressed as ( v = 7x^{3} ). the volume is to be calculated with an error of no more than 1% of the true value. find approximately the greatest error that can be tolerated in the measurement of x, expressed as a percentage of x.\n\nthe greatest tolerated error in the measurement of x is (square%).\n(type an integer or a simplified fraction.)

the volume of a solid can be expressed as ( v = 7x^{3} ). the volume is to be calculated with an error of no more than 1% of the true value. find approximately the greatest error that can be tolerated in the measurement of x, expressed as a percentage of x.\n\nthe greatest tolerated error in the measurement of x is (square%).\n(type an integer or a simplified fraction.)

Answer

Explanation:

Step1: Differentiate the volume formula

Given ( V = 7x^{3} ), using the power rule ( (x^{n})^\prime=nx^{n - 1} ), we get ( dV=21x^{2}dx ).

Step2: Find the relative error formula

The relative error in ( V ) is ( \frac{dV}{V} ), and the relative error in ( x ) is ( \frac{dx}{x} ). Substitute ( V = 7x^{3} ) and ( dV = 21x^{2}dx ) into ( \frac{dV}{V} ): [ \frac{dV}{V}=\frac{21x^{2}dx}{7x^{3}} = 3\frac{dx}{x} ]

Step3: Solve for ( \frac{dx}{x} )

We know that ( \frac{dV}{V}\leq 0.01 ). Since ( \frac{dV}{V}=3\frac{dx}{x} ), then ( 3\frac{dx}{x}\leq 0.01 ). Solving for ( \frac{dx}{x} ), we get ( \frac{dx}{x}\leq\frac{0.01}{3}=\frac{1}{300}\approx 0.0033).

Answer:

(\frac{1}{3}%)