Respuesta :
(a) Concerning the coefficients of the equation for the volume of the eggplant ( [tex]V(ft^{3}) = 3.53\times 10^{-2}\cdot e^{2\cdot t^{2}}[/tex]), the units of the coefficients with values [tex]3.53 \times 10^{-2}[/tex] and 2 are cubic feet and [tex]\frac{1}{h^{2}}[/tex], respectively.
(b) Regarding the determination of the coefficients by linear regression, the scientist must plot the square of time versus the natural logarithm of the volume and he would have obtained that the intercept is the natural logarithm of constant A and the slope is constant B.
(c) After the solicitation of the European distributor of the film, the new formula is [tex]V' = 9.884\times 10^{-4}\cdot e^{1.543\times 10^{-7}\cdot t'^{2}}[/tex], where volume is in cubic meters and time is in seconds.
Note - The statement reports typing mistakes. We present the correct statement below:
The climatic moment in the film "The Eggplant that Ate New Jersey" comes when the brilliant young scientist announces his discovery of the equation for the volume of the eggplant: [tex]V(ft^{3}) = 3.53\times 10^{-2}\cdot e^{2\cdot t^{2}}[/tex], where [tex]t[/tex] is the time in hours from the moment the vampire injected the eggplant with a solution prepared from the blood of the beautiful dental hygienist.
(a) What are the units of [tex]3.53\times 10^{-2}[/tex] and [tex]2[/tex]?
(b) The scientist obtained the formula by measuring [tex]V[/tex] and [tex]t[/tex] and determining the coefficients by linear regression. What would he have plotted versus what on what kind of coordinates? What would he have obtained as the slope and intercept of his plot?
(c) The European distributor of the film insists that the formula must be given for the volume in cubic meters ([tex]m^{3}[/tex]) as a function of time, in seconds. Derive the formula.
(a) In this exercise, we must make use of principles for dimensional analysis. We know that the equation for the volume of the eggplant is represented by:
[tex]V(ft^{3}) = 3.53\times 10^{-2}\cdot e^{2\cdot t^{2}}[/tex] (1)
Dimensionally speaking, we have the following case:
[tex][L]^{3} = [A]\cdot e^{[B]\cdot [t]^{2}}[/tex] (2)
By direct inspection and some algebraic handling, we conclude that:
[tex][A] = [L]^{3}[/tex]
[tex][B] = [t]^{-2}[/tex]
Therefore, the units of [tex]3.53 \times 10^{-2}[/tex] and 2 are cubic feet and [tex]\frac{1}{h^{2}}[/tex], respectively.
(b) The curve fit for the exponential expression used by the scientist is of the form:
[tex]V = A\cdot e^{B\cdot t^{2}}[/tex] (3)
If we apply logarithms on both sides of the expression above, then we have the following form:
[tex]\ln V = \ln A\cdot e^{B\cdot t^{2}}[/tex]
[tex]\ln V = \ln A + B\cdot t^{2}[/tex] (4)
The scientist must plot the square of time versus the natural logarithm of the volume and he would have obtained that the intercept is the natural logarithm of constant A and the slope is constant B.
(c) An hour equals 3600 seconds and a cubic feet equals 0.028 cubic meters. Hence, we can adapt (1) to fulfill the requirements of the European film distributor by using the following conversions:
[tex]t' = 3600\cdot t[/tex] (5)
[tex]V' = 0.028\cdot V[/tex] (6)
Where:
- [tex]t[/tex] - Time, in hours.
- [tex]t'[/tex] - Time, in seconds.
- [tex]V[/tex] - Volume, in cubic feet.
- [tex]V'[/tex] - Volume, in cubic meters.
By applying (5) and (6) in (1), we have the resulting formula:
[tex]V' = 0.028\cdot [3.53\times 10^{-2}\cdot e^{2\cdot \left(\frac{t'}{3600} \right)^{2}}][/tex]
[tex]V' = 9.884\times 10^{-4}\cdot e^{1.543\times 10^{-7}\cdot t'^{2}}[/tex]
The new formula is [tex]V' = 9.884\times 10^{-4}\cdot e^{1.543\times 10^{-7}\cdot t'^{2}}[/tex], where volume is in cubic meters and time is in seconds.
We kindly invite to check this question on exponential functions: https://brainly.com/question/2456547
After a small online search, I've found that the volume equation is:
[tex]V[ft^3] = 3.53 \cdot10^{-2} \cdot exp(2 \cdot t^2)[/tex]
a) We want to see what are the units of the first factor:
[tex]3.53 \cdot10^{-2}[/tex]
You can see that the volume is in cube feet, and the only part in the equation that can have units is this one, so the units of this factor should be cube feet.
b) On the vertical coordinate he would see the volume and on the horizontal coordinate he would see the time.
The graph of this function is an increasing exponential function, that grows slow at the beginning and then grows faster.
c) Remember that the term with the units is:
[tex]3.53 \cdot10^{-2}[/tex]
So we should write this as:
[tex]3.53 \cdot10^{-2} ft^3[/tex]
Now we know that:
[tex]1m^3 = 35.3 ft^3\\\\1 = \frac{1 m^3}{35.3 ft^3}[/tex]
Now we can multiply our measure by this to get:
[tex]3.53 \cdot10^{-2} ft^3 = ( \frac{1 m^3}{35.3 ft^3})*(3.53 \cdot10^{-2} ft^3) = 0.1 m^3[/tex]
Then the equation is:
[tex]V = 0.1m^3\cdot exp(2 \cdot t^2)[/tex]
If you want to learn more you can read:
https://brainly.com/question/172096
