a) The equation [tex]V(m) = 8400 - 560\cdot m[/tex] represents the discharge of the Grand Prismatic Spring.
b) The discharge of the Grand Prismatic Spring takes 15 minutes.
c) The remaining quantity after 7 minutes is 4480 gallons.
d) The hardness of cardboard is 2.
How to apply concepts of linear functions
a) In this case, the volume of water ([tex]V[/tex]), in gallons, decreases linearly in time ([tex]m[/tex]), in minutes. Then, we could model the situation by using the following expression:
[tex]V(m) = V_{o} - \dot V \cdot m[/tex] (1)
Where:
- [tex]V_{o}[/tex] - Initial volume, in gallons.
- [tex]\dot V[/tex] - Discharge rate, in gallons per minute.
If we know that [tex]V_{o} = 8400\,gal[/tex] and [tex]\dot V = 560\,\frac{gal}{min}[/tex], then the volume of the Grand Prismatic Spring is represented by this expression:
[tex]V(m) = 8400 - 560\cdot m[/tex] (2)
The equation [tex]V(m) = 8400 - 560\cdot m[/tex] represents the discharge of the Grand Prismatic Spring. [tex]\blacksquare[/tex]
b) By (1) and [tex]V(m) = 0[/tex] we find the time required for discharge:
[tex]m = \frac{0 - 8400}{-560}[/tex]
[tex]m = 15\,min[/tex]
The discharge of the Grand Prismatic Spring takes 15 minutes. [tex]\blacksquare[/tex]
c) The quantity of gallons remaining is found by evaluating the function for [tex]m = 7[/tex]:
[tex]V(7) = 8400 - 560\cdot (7)[/tex]
[tex]V(7) = 4480\,gal[/tex]
The remaining quantity after 7 minutes is 4480 gallons. [tex]\blacksquare[/tex]
d) According to Mohs scale, the hardness of feldspar is 6. The statement indicates the following relationship, as there is a constant relationship between both hardnesses:
[tex]y = 3\cdot x[/tex] (3)
Where:
- [tex]x[/tex] - Hardness of cardboard.
- [tex]y[/tex] - Hardness of feldspar.
If we know that [tex]y = 6[/tex], then the hardness of cardboard is:
[tex]x = \frac{6}{3}[/tex]
[tex]x = 2[/tex]
The hardness of cardboard is 2. [tex]\blacksquare[/tex]
To learn more on linear functions, we kindly invite to check this verified question: https://brainly.com/question/17058347
