<h2>9.</h2><h3>Given</h3>
<h3>Find</h3>
- linear approximation to the volume when the radius increases 0.4 cm
<h3>Solution</h3>
The equation for volume of a sphere is
... V = (4/3)π·r³
Differentiating gives
... dV = 4π·r²·dr
Filling in the given numbers gives
... change in volume ≈ 4π·(15 cm)²·(0.4 cm)
... = 360π cm³ ≈ 1130.97 cm³ . . . . . . volume of layer 4mm thick
<h2>11.</h2><h3>Given</h3>
- an x by x by 2x cuboid with surface area 129.6 cm²
- rate of change of x is 0.01 cm/s
<h3>Find</h3>
<h3>Solution</h3>
The area is that of two cubes of dimension x joined together. The area of each such cube is 6x², but the two joined faces don't count in the external surface area. Thus the area of the cuboid is 10x².
The volume of the cuboid is that of two cubes joined, so is 2x³. Then the rate of change of volume is
... dV/dt = (d/dt)(2x³) = 6x²·dx/dt
We know x² = A/10, where A is the area of the cuboid, so the rate of change of volume is ...
... dV/dt = (6/10)A·dx/dt = 0.6·(129.6 cm²)(0.01 cm/s)
... dV/dt = 0.7776 cm³/s
These are two separate problems: in the first we will have to substitute in a new value for x into the original equation and in the second we will manipulate the preexisting equation for f(x).
To begin, we will sub in f(x/3). To do this, we will substitute each variable x in the equation (in this case there is only one) with x/3, and then simplify the resulting equation.
f(x) = 6x - 18
f(x/3) = 6(x/3) - 18
To simplify, we should distribute the 6 on the right side of the equation.
f(x/3) = 6x/3 - 18
Now, we can divide the first term on the right side to finalize our simplification.
f(x/3) = 2x -18
Secondly, we are asked to find f(x)/3. To do this, we will take our original value for f(x), and then simplify divide that entire function by 3.
f(x) = 6x - 18
f(x)/3 = (6x-18)/3
This means that we must divide each term of the binomial by 3, so we are really computing
f(x)/3 = 6x/3 - 18/3
We can simplify by dividing both of the terms.
f(x)/3 = 2x - 6
Therefore, your answer is that f(x/3) = 2x - 18, but f(x)/3 = 2x - 6. It is important to recognize that these are two similar, yet different, answers.
Hope this helps!
Answer:
30 pounds is 13.6078 kilograms.
13 x 16 = 208.
Mary's dog should be given 208 mg of medicine.
Step-by-step explanation:
Use the form (a + b)(a - b) = a^2 - b^2:
(2x^4 + 3x^3)(2x^4 - 3x^3) = 4x^8 - 9x^6.
