Answer:

Step-by-step explanation:
Consider the revenue function given by
. We want to find the values of each of the variables such that the gradient( i.e the first partial derivatives of the function) is 0. Then, we have the following (the explicit calculations of both derivatives are omitted).


From the first equation, we get,
.If we replace that in the second equation, we get

From where we get that
. If we replace that in the first equation, we get

So, the critical point is
. We must check that it is a maximum. To do so, we will use the Hessian criteria. To do so, we must calculate the second derivatives and the crossed derivatives and check if the criteria is fulfilled in order for it to be a maximum. We get that


We have the following matrix,
.
Recall that the Hessian criteria says that, for the point to be a maximum, the determinant of the whole matrix should be positive and the element of the matrix that is in the upper left corner should be negative. Note that the determinant of the matrix is
and that -10<0. Hence, the criteria is fulfilled and the critical point is a maximum
Answer:
Sue would have taken 15 hours to sew all the costumes.
Step-by-step explanation:
In 10 hours, Sue sewed 5 costumes (1 every 2 hours).
Sue= 5 costumes
Sue sewed in 10 hours, twice as many costumes as Anne in 16 hours. Then 5 is twice costumes than Anne produced. Therefore, Anne sewed 2.5 costumes.
Anne: 2.5 costumes
In total they had to sew 5+2.5= 7.5 costumes. Then Sue would have taken 7.5*2= 15 hours to sew them all.
30 + 70 + D = 180
100 + D = 180
D = 80°
Answer:
Explanation:
You need to use derivatives which is an advanced concept used in calculus.
<u>1. Write the equation for the volume of the cone:</u>

<u />
<u>2. Find the relation between the radius and the height:</u>
- r = diameter/2 = 5m/2 = 2.5m
<u>3. Filling the tank:</u>
Call y the height of water and x the horizontal distance from the axis of symmetry of the cone to the wall for the surface of water, when the cone is being filled.
The ratio x/y is the same r/h
The volume of water inside the cone is:


<u>4. Find the derivative of the volume of water with respect to time:</u>

<u>5. Find x² when the volume of water is 8π m³:</u>
m²
<u>6. Solve for dx/dt:</u>


<u />
<u>7. Find dh/dt:</u>
From y/x = h/r = 2.08:

That is the rate at which the water level is rising when there is 8π m³ of water.