Answer
2.93 inches
Work Step by Step
To solve this problem, we need to first find the volume of the 1.000 kg mass that is to be cut from the steel bar. We can then use this volume to find the length of the section of the bar needed.
The equilateral triangular cross-section of the steel bar has sides of 2.50 in, so its area can be found using the formula for the area of an equilateral triangle:
$A = \frac{\sqrt 3}{4} * s^2$
where s is the length of the side of the triangle. Substituting s = 2.50 in gives us:
$A = \frac{\sqrt 3}{4} * (2.50 in)^2 ≈ 2.706 in^2$
To find the volume of the 1.000 kg mass that is to be cut from the steel bar, we can use the density of the steel:
$density = mass / volume$
Rearranging this formula, we get:
$volume = mass / density$
Substituting the given values, we get:
$volume = 1.000 kg / (7.70 g/cm^3) ≈ 0.12987 L$
To convert this volume to cubic inches, we can use the conversion factor:
$1 L = 61.0237 in^3$
So, the volume of the 1.000 kg mass is:
$0.12987 L * 61.0237 in^3/L ≈ 7.925 in^3$
Finally, we can use the formula for the volume of a triangular prism to find the length of the section of the bar needed:
$V = A * l$
where A is the area of the triangular cross-section, l is the length of the section of the bar needed, and V is the volume of the 1.000 kg mass. Substituting the given values, we get:
$7.925in^3 = 2.706 in^2 * l$
Solving for l gives us:
$l = (7.925in^3) / (2.706 in^2) ≈ 2.929 in ≈ 2.93 in$
Therefore, the section of the bar needs to be approximately 2.93 inches long.
