VIPSolutionsGive step-by-step solution with explanation and final answer: Recall that profit equals revenue minus cost. For the revenue and cost functions shown, answer parts ( a ) through ( e ) below. R ( x ) = 4 2 0 x - 2 x 2 and C ( x ) = 1 0 0 x 9 8 0 0 , with 0 ≤ x ≤ 1 5 0 a ) Find the break - even point. The break - even point ( s ) is ( are ) x = ◻ . Type an integer or decimal rounded to one decimal place as needed. Use a comma to separate answers as needed. )33 gee Ha §17 |e 13 8 i : i : i I oll i l wr . i
Question:
Give step-by-step solution with explanation and final answer:
Recall that profit equals revenue minus cost. For the revenue and cost functions shown, answer parts
(
a
)
through
(
e
)
below.
R
(
x
)
=
4
2
0
x
-
2
x
2
and
C
(
x
)
=
1
0
0
x
9
8
0
0
,
with
0
≤
x
≤
1
5
0
a
)
Find the break
-
even point.
The break
-
even point
(
s
)
is
(
are
)
x
=
◻
.
Type an integer or decimal rounded to one decimal place as needed. Use a comma to separate answers as needed.
)
33 gee
Ha
§17 |e
13 8
i
: i
: i
I oll
i l
wr
. i
This is AI generated Response.
Asked by: anil
Created at: 2025-09-29 08:41:08
Asked by: anil
Created at: 2025-09-29 08:41:08
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Let's solve the break-even point step by step.
### **Step 1: Set Revenue Equal to Cost**
The break-even point occurs when revenue equals cost:
\[
R(x) = C(x)
\]
Given:
\[
R(x) = 420x 2x^2
\]
\[
C(x) = 100x + 980
\]
Set them equal to each other:
\[
420x - 2x^2 = 100x + 980
\]
---
### **Step 2: Rearrange the Equation**
Bring all terms to one side:
\[
420x - 2x^2 - 100x - 980 =
\]
\[
(420x - 100x) - 2x^2 - 980 =
\]
\[
320x - 2x^2 - 980 =
\]
Multiply both sides by -1 to make the quadratic term positive:
\[
2x^2 - 320x + 980 =
\]
---
### **Step 3: Solve the Quadratic Equation**
This is a quadratic equation in the form \( ax^2 + bx + c = \):
- \( a = 2 \)
- \( b = -320 \)
- \( c = 980 \)
Use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Plug in the values:
\[
x = \frac{-(-320) \pm \sqrt{(-320)^2 - 4 \cdot 2 \cdot 980}}{2 \cdot 2}
\]
\[
x = \frac{320 \pm \sqrt{102400 - 78400}}{4}
\]
\[
x = \frac{320 \pm \sqrt{24000}}{4}
\]
---
### **Step 4: Simplify**
\[
\sqrt{24000} \approx 154.9
\]
So,
\[
x_1 = \frac{320 + 154.9}{4} = \frac{474.9}{4} = 118.7
\]
\[
x_2 = \frac{320 - 154.9}{4} = \frac{165.1}{4} = 41.3
\]
---
### **Step 5: State the Break-Even Points**
**Final Answer:**
\[
\boxed{41.3, 118.7}
\]
- **These are the break-even points, rounded to one decimal place.**
- Both values are within the given range \( \leq x \leq 150\).
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