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The profit from the production and sale of specialty golf hats is given by the function [tex]\( P(x) = 20x - 6000 \)[/tex] where [tex]\( x \)[/tex] is the number of hats produced and sold.

(a) Producing and selling how many hats will give a profit of [tex]$8000?

(b) How many hats must be produced and sold to avoid a loss?

(a) Producing and selling \( \square \) hats will give a profit of $[/tex]8000.

Sagot :

GinnyAnswer
Sure! Let's break down the problem step by step.

### Part (a)
To find the number of hats needed to achieve a profit of [tex]$8000, we need to set \( P(x) \) equal to $[/tex]8000 and solve for [tex]\( x \)[/tex]. The given profit function is:

[tex]\[ P(x) = 20x - 6000 \][/tex]

We set this equal to [tex]$8000: \[ 20x - 6000 = 8000 \] Now, solve for \( x \): 1. Add 6000 to both sides of the equation: \[ 20x = 8000 + 6000 \] \[ 20x = 14000 \] 2. Divide both sides by 20: \[ x = \frac{14000}{20} \] \[ x = 700 \] So, producing and selling 700 hats will give a profit of $[/tex]8000.

### Part (b)
To avoid a loss, the profit must be at least zero. This means we need to find the number of hats [tex]\( x \)[/tex] for which [tex]\( P(x) = 0 \)[/tex]. Set the profit function equal to zero:

[tex]\[ 20x - 6000 = 0 \][/tex]

Now, solve for [tex]\( x \)[/tex]:

1. Add 6000 to both sides of the equation:

[tex]\[ 20x = 6000 \][/tex]

2. Divide both sides by 20:

[tex]\[ x = \frac{6000}{20} \][/tex]
[tex]\[ x = 300 \][/tex]

So, producing and selling at least 300 hats will avoid a loss.

### Summary
(a) Producing and selling 700 hats will give a profit of $8000.
(b) Producing and selling at least 300 hats will avoid a loss.
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