nment A company charges $$ \$ 20 $$ to make one monogrammed shirt, but reduces this cost by $$ \$ 0.10 $$ per shirt for each additional shirt ordered up to 100 shirts. If the of an order is $$ \$ 846 $$, how many shirts were ordered? If the cost of an order is $$ \$ 846.2 .145 $$, $$ \square $$ shirts were ordered. (Simplify your answer.)

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We let $$ x $$ be the number of shirts ordered. The company’s pricing rule is that the cost to make one shirt is reduced by $$\$0.10$$ for every additional shirt ordered. In other words, the cost per shirt when $$ x $$ shirts are ordered becomes
$$
20 - 0.10(x-1).
$$
Simplify the expression:
$$
20 - 0.10(x-1)=20-0.10x+0.10=20.10-0.10x.
$$

The total cost of the order, $$ C $$, is the number of shirts times the cost per shirt:
$$
C = x(20.10-0.10x).
$$

For the first part of the problem the total cost is given as $$\$846$$. Thus the equation is:
$$
x(20.10-0.10x)=846.
$$

To clear decimals, multiply both sides by 10:
$$
x(201- x)=8460.
$$

Expanding, we obtain:
$$
201x-x^2=8460.
$$

Writing in standard quadratic form:
$$
x^2-201x+8460=0.
$$

Next, we compute the discriminant:
$$
\Delta = 201^2-4\cdot 1\cdot8460.
$$
Calculate:
$$
201^2=40401,
$$
$$
4\cdot8460=33840,
$$
so
$$
\Delta=40401-33840=6561.
$$
Since
$$
\sqrt{6561}=81,
$$
the solutions for $$ x $$ are given by:
$$
x=\frac{201\pm 81}{2}.
$$
Thus,
$$
x=\frac{201+81}{2}=\frac{282}{2}=141 \quad \text{or} \quad x=\frac{201-81}{2}=\frac{120}{2}=60.
$$

Because the discount applies only for orders up to 100 shirts, the value $$ x=141 $$ is not allowable. Therefore, the number of shirts ordered is
$$
\boxed{60}.
$$
60 shirts were ordered.

nment
A company charges to make one monogrammed shirt, but reduces this cost by per shirt for each additional shirt ordered up to 100 shirts. If the
of an order is , how many shirts were ordered?
If the cost of an order is , shirts were ordered. (Simplify your answer.)

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nment A company charges to make one monogrammed shirt, but reduces this cost by per shirt for each additional shirt ordered up to 100 shirts. If the of an order is , how many shirts were ordered? If the cost of an order is , shirts were ordered. (Simplify your answer.)