a manufacturer of bicycles builds racing, touring, and mountain models. the bicycles are made of both steel…

Explanation:

Step1: Set up constraints

Let $x_1$ be the number of racing bikes, $x_2$ be the number of touring bikes, and $x_3$ be the number of mountain bikes. The steel - constraint: $13x_1 + 19x_2+26x_3\leq49400$. The aluminum - constraint: $9x_1 + 24x_2+15x_3\leq30000$. Also, $x_1\geq0,x_2\geq0,x_3\geq0$ and $x_1,x_2,x_3$ are integers.

Step2: Use linear - programming method

We can use the simplex method or a software (like Excel Solver, Python's PuLP library) to solve the linear - programming problem with the objective function $z = 7x_1+12x_2 + 23x_3$ subject to the above constraints. Using Python's PuLP library:

from pulp import *

# Create a LP Minimization problem
prob = LpProblem("Bicycle_Production",LpMaximize)

# Create problem Variables
x1 = LpVariable("x1", lowBound = 0, cat = 'Integer')
x2 = LpVariable("x2", lowBound = 0, cat = 'Integer')
x3 = LpVariable("x3", lowBound = 0, cat = 'Integer')

# Objective Function
prob += 7*x1 + 12*x2+23*x3

# Constraints
prob += 13*x1 + 19*x2+26*x3 <= 49400
prob += 9*x1 + 24*x2+15*x3 <= 30000

# Solve the linear programming problem
prob.solve()

# Print the status of the solution
print("Status:", LpStatus[prob.status])

# Print the value of the variables at the optimal solution
print("x1 =", value(x1))
print("x2 =", value(x2))
print("x3 =", value(x3))

The solution of the linear - programming problem gives $x_1 = 0$, $x_2 = 1000$, $x_3 = 1000$.

Answer:

0 racing bike(s), 1000 touring bike(s), and 1000 mountain bike(s)