Learn Before
Maximum Load Supported by a Beam: Direct Variation with the Square
Problem: The maximum load a beam can support varies directly with the square of the diagonal of the beam's cross-section. A beam with a 4-inch diagonal supports a maximum load of 75 pounds. (a) Write the equation relating maximum load to the diagonal. (b) What maximum load can a beam with an 8-inch diagonal support?
Part (a) — Find the equation.
Let = maximum load and = diagonal of the cross-section. Because varies directly with the square of , write the direct variation formula:
Substitute the known values and :
Divide both sides by 16 to solve for :
Substitute back into the formula:
Part (b) — Find when .
Substitute into the equation:
A beam with an 8-inch diagonal can support a maximum load of 300 pounds. Notice that doubling the diagonal from 4 inches to 8 inches quadruples the maximum load from 75 to 300 pounds, which reflects the squared relationship in the variation formula.
0
1
Tags
OpenStax
Elementary Algebra @ OpenStax
Ch.8 Rational Expressions and Equations - Elementary Algebra @ OpenStax
Algebra
Math
Prealgebra
Related
Direct Variation with the Square of a Variable
Maximum Load Supported by a Beam: Direct Variation with the Square
A payroll clerk uses the direct variation equation S = 18h to calculate an employee's salary (S) based on hours worked (h). In this equation, the fixed hourly rate of 18 is mathematically known as the constant of ____.
A freelance consultant charges a fixed hourly rate for their services. If the total fee (F) varies directly with the number of hours worked (h), which equation represents this relationship using 'k' as the constant of variation?
A logistics manager tracks fleet operations and notes that the total fuel consumption varies directly with the total distance driven. Match the following components of the direct variation model to their specific descriptions in this professional context.
In a direct variation relationship, such as a company's total revenue varying directly with the number of units sold, doubling the number of units sold will result in doubling the total revenue.
Direct Variation in Production Planning
A project manager is developing a cost-estimation model where the total service fee varies directly with the number of hours worked. Arrange the following steps in the correct chronological order to establish the specific mathematical equation for this relationship.
Direct Variation in Professional Billing
Defining Direct Variation for Professional Contexts
An operations manager is modeling the relationship between electricity usage and machine run-time. If the usage varies directly with the run-time, what is the effect on the total electricity usage if the machine run-time is exactly tripled?
A project lead is creating a visual report showing that the total cost of materials varies directly with the number of units produced. When this direct variation relationship is graphed on a coordinate plane, which specific point must the line pass through to represent the scenario where no units are produced?
Solving a Direct Variation Application: Calories Burned Exercising
Solving a Direct Variation Application: Distance Traveled by a Train
Learn After
A construction supervisor needs to determine the load capacity of various support beams. If the maximum load (L) a beam can support varies directly with the square of the diagonal (c) of its cross-section, which equation correctly models this relationship using a constant of variation (k)?
A construction supervisor is using a formula to determine the load limits for support beams. If the maximum load (L) varies directly with the square of the diagonal (c), the relationship is expressed by the equation L = kc^n. In this equation, the value of the exponent 'n' is ____.
A structural safety officer is reviewing the relationship between beam dimensions and weight limits. If the maximum load a beam can support varies directly with the square of its diagonal, then doubling the diagonal length will result in the beam supporting exactly four times as much weight.
A structural safety inspector is reviewing the load-bearing specifications for support beams. To accurately model the relationship where the maximum load () varies directly with the square of the beam's diagonal (), match each component of the formula to the physical or mathematical property it represents.
A structural inspector is establishing a weight-limit model for a series of support beams. The maximum load (L) is known to vary directly with the square of the beam's diagonal cross-section (c). Arrange the following steps in the correct order to determine the specific equation for this relationship based on an initial test measurement.
Structural Load-Bearing Equation Identification
Structural Integrity Standards
Structural Safety Standards: Beam Load Variation
A structural safety manual provides the formula to calculate the maximum weight limit () for a specific series of support beams based on the diagonal of their cross-section (). Based on the algebraic structure of this equation, which of the following specifically describes the relationship between the load and the diagonal?
A construction technician is using the principle of direct variation with the square to model the weight limits of support beams. After being given a known test result (the load capacity for one specific diagonal), which value must the technician calculate first to establish the general mathematical model for all beams of that specific type?