For the derivative tests method, assume that the sphere is centered at the origin, and consider the
circular projection of the sphere onto the xy-plane. An inscribed rectangular box is uniquely determined
1
by the xy-coordinate of its corner in the first octant, so we can compute the z coordinate of this corner
by
x2+y2+z2=r2 =⇒z= r2−(x2+y2).
Then the volume of a box with this coordinate for the corner is given by
V = (2x)(2y)(2z) = 8xy r2 − (x2 + y2),
and we need only maximize this on the domain x2 + y2 ≤ r2. Notice that the volume is zero on the
boundary of this domain, so we need only consider critical points contained inside the domain in order
to carry this optimization out.
For the method of Lagrange multipliers, we optimize V(x,y,z) = 8xyz subject to the constraint
x2 + y2 + z2 = r2<span>. </span>
We must look at this question in steps
The first half of the journey is travelled at 40 km/h
Half of 100km is 50 km
Using the formula
Distance = Speed x Time
Speed = Distance / Time
Time = Distance / Speed
We can work out the time:
50km / 40km/h = 1.25 hours
Next we look at the second half of the journey
50km at 80km/h
50km / 80km/h = 0.625 hours
Add together both times to work out how long the entire journey took
1.25 + 0.625 = 1.875 hours
Using the Speed formula from before
Speed = 100km / 1.875 =
53 1/3 km/h or 53.3 recurring km/h
Answer:
x=10
Step-by-step explanation:
