Answer:
The maximum volume is 1417.87 ![inch^3](https://tex.z-dn.net/?f=inch%5E3)
Explanation:
<u>Optimization Using Derivatives</u>
We have a 24x30 inch piece of metal and we need to make a rectangular box by cutting a square from each corner of the piece and bending up the sides. The width of the piece is 24 inches and its length is 30 inches
When we cut a square of each corner of side x, the base of the box (after bending up the sides) will be (24-2x) and (30-2x), width and length respectively. The volume of the box is
![V=(24-2x)(30-2x)x](https://tex.z-dn.net/?f=V%3D%2824-2x%29%2830-2x%29x)
Operating
![V=4x^3-108x^2+720x](https://tex.z-dn.net/?f=V%3D4x%5E3-108x%5E2%2B720x)
To find the maximum value of V, we compute the first derivative and equate it to zero
![V'=12x^2-216x+720=0](https://tex.z-dn.net/?f=V%27%3D12x%5E2-216x%2B720%3D0)
Simplifying by 12
![x^2-18x+60=0](https://tex.z-dn.net/?f=x%5E2-18x%2B60%3D0)
Completing squares
![x^2-18x+81-81+60=0](https://tex.z-dn.net/?f=x%5E2-18x%2B81-81%2B60%3D0)
![(x-9)^2=21](https://tex.z-dn.net/?f=%28x-9%29%5E2%3D21)
We have two values for x
![x=9+\sqrt{21}=13.58\ inch](https://tex.z-dn.net/?f=x%3D9%2B%5Csqrt%7B21%7D%3D13.58%5C%20inch)
![x=9-\sqrt{21}=4.42\ inch](https://tex.z-dn.net/?f=x%3D9-%5Csqrt%7B21%7D%3D4.42%5C%20inch)
The first value is not feasible because it will produce a negative width (24-2(13.58))=-6.16
We'll keep only the solution
![x=4.42\ inch](https://tex.z-dn.net/?f=x%3D4.42%5C%20inch)
The width is
![w=(24-2(4.42))=15.16\ inch](https://tex.z-dn.net/?f=w%3D%2824-2%284.42%29%29%3D15.16%5C%20inch)
The length is
![l=(30-2(4.42))=21.16\ inch](https://tex.z-dn.net/?f=l%3D%2830-2%284.42%29%29%3D21.16%5C%20inch)
And the height
![x=4.42\ inch](https://tex.z-dn.net/?f=x%3D4.42%5C%20inch)
The maximum volume is
![V=(15.16)(21.16)(4.42)=1417.87\ inch^3](https://tex.z-dn.net/?f=V%3D%2815.16%29%2821.16%29%284.42%29%3D1417.87%5C%20inch%5E3)
Answer:
alcohol thermometer is preferred for a very cold region. Alcohol thermometer is preferred for a very cold region because its freezing point of alcohol is - 117°C. So, it can measure the temperature of the very cold region.
Answer:
The mechanical advantage of the pulley is 1.
Explanation:
The mechanical advantage of the pulley is given as
![MA=\frac{F}{f}](https://tex.z-dn.net/?f=MA%3D%5Cfrac%7BF%7D%7Bf%7D)
Here, MA is represent for the mechanical advantage of the pulley, F is load for lift and f applied force.
Given F =21 N and f= 21 N.
Substitute the given values, we get
![MA=\frac{21N}{21N}](https://tex.z-dn.net/?f=MA%3D%5Cfrac%7B21N%7D%7B21N%7D)
MA = 1.
Thus, the mechanical advantage of the pulley is 1.
Answer:
1.97 x 10^8 m/s
Explanation:
refractive index of crown glass with respect to air, n = 1.52
speed of light in air, c = 3 x 10^8 m/s
Let v be the speed of light in crown glass.
By use of the definition of refractive index
![n = \frac{c}{v}](https://tex.z-dn.net/?f=n%20%3D%20%5Cfrac%7Bc%7D%7Bv%7D)
where, n be the refractive index of crown glass, c be the speed of light in vacuum and v be the speed of light in crown glass
![1.52 = \frac{3 \times 10^{8}}{v}](https://tex.z-dn.net/?f=1.52%20%3D%20%5Cfrac%7B3%20%5Ctimes%2010%5E%7B8%7D%7D%7Bv%7D)
![v = \frac{3 \times 10^{8}}{1.52}](https://tex.z-dn.net/?f=v%20%3D%20%5Cfrac%7B3%20%5Ctimes%2010%5E%7B8%7D%7D%7B1.52%7D)
v = 1.97 x 10^8 m/s
Thus, the speed of light in crown glass is 1.97 x 10^8 m/s.