Answer:
17) MC(x) = 35 − 12/x²
18) R(x) = -0.05x² + 80x
Step-by-step explanation:
17) The marginal average cost function (MC) is the derivative of the average cost function (AC).
AC(x) = C(x) / x
MC(x) = d/dx AC(x)
First, find the average cost function:
AC(x) = C(x) / x
AC(x) = (5x + 3)(7x + 4) / x
AC(x) = (35x² + 41x + 12) / x
AC(x) = 35x + 41 + 12/x
Now find the marginal average cost function:
MC(x) = d/dx AC(x)
MC(x) = 35 − 12/x²
18) x is the demand, and p(x) is the price at that demand. Assuming the equation is linear, let's use the points to find the slope:
m = (40 − 50) / (800 − 600)
m = -0.05
Use point-slope form to find the equation of the line:
p(x) − 50 = -0.05 (x − 600)
p(x) − 50 = -0.05x + 30
p(x) = -0.05x + 80
The revenue is the product of price and demand:
R(x) = x p(x)
R(x) = x (-0.05x + 80)
R(x) = -0.05x² + 80x
17)
The marginal average cost function is [tex]C'(x)=70x+41[/tex]
"If C(x) is the total cost of producing x items, then C'(x) is called the marginal cost."
For given question,
We have been given the total cost to produce x units of paint
C(x) = (5x + 3)(7x + 4)
We need find the marginal average cost function.
⇒ C(x) = (5x + 3)(7x + 4)
⇒ C(x) = 5x(7x + 4) + 3(7x + 4)
⇒ C(x) = 35x² + 20x + 21x +12
⇒ C(x) = 35x² + 41x + 12
so, the marginal average cost function would be,
[tex]C'(x) \\\\=\frac{d}{dx}C(x)\\\\ =\frac{d}{dx} (35x^2+41x+12)\\\\=\frac{d}{dx} (35x^2)+\frac{d}{dx}(41x)+ \frac{d}{dx}(12)\\\\=70x+41+0\\\\=70x+41[/tex]
Therefore, the marginal average cost function is [tex]C'(x)=70x+41[/tex]
Learn more about the marginal average cost function here:
https://brainly.com/question/16912525
18)
The revenue equation in terms of the demand x is [tex]R(x)=-0.05x^2+80x[/tex]
p = mq + b
where p is the price,
m is the slope
q is the number of quantity
d is all factors affecting price other than price
"Revenue = demand equation* Quantity Sold"
For given question,
Let x is the quantity which is demanded
P is the price per blender
Now we are given that x = 600 blenders are demanded at price P =$50 per blender.
So, the corresponding point would be given as (600, 50)
Also, x = 800 blenders are demanded at price P =$40 per blender.
So, the corresponding point would be given as (800, 40)
The equation of the line passing through these points would be,
[tex]\Rightarrow \frac{P-50}{40-50}= \frac{x-600}{800-600}\\\\ \Rightarrow \frac{P-50}{-10} =\frac{x-600}{200}\\\\ \Rightarrow \frac{200}{-10}(P-50)=x-600\\\\ \Rightarrow -20(P-50)=x-600\\\\ \Rightarrow -20P+1000=x-600\\\\ \Rightarrow -20P=x-600-1000\\\\ \Rightarrow -20P=x-1600\\\\ \Rightarrow P=\frac{1}{20}x+80\\\\ \Rightarrow P=-0.05x+80[/tex]
So, the demand equation is [tex]P=-0.05x+80[/tex]
Now, the revenue equation in terms of the demand x would be,
[tex]\Rightarrow R(x)=P.x\\\\\Rightarrow R(x)= (-0.05x+80)x\\\\\Rightarrow R(x)=-0.05x^2+80x[/tex]
Therefore, the revenue equation in terms of the demand x is [tex]R(x)=-0.05x^2+80x[/tex]
Learn more about the revenue equation here:
https://brainly.com/question/13420722
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