Answer:
Therefore the rate of change of area
square inches/ s
Step-by-step explanation:
Given that,
A rectangle is growing such that the length of rectangle is(5t+4) and its height is t⁴.
Where t is in second and dimensions are in inches.
The area of a rectangle is = length× height
Therefore the area of the rectangle is
A(t) = (5t+4) t⁴
⇒A(t) = t⁴(5t+4)
To find the rate change of area we need to find out the first order derivative of the area.
Rules:
![(1)\frac{dx^n}{dx} = nx^{n-1}](https://tex.z-dn.net/?f=%281%29%5Cfrac%7Bdx%5En%7D%7Bdx%7D%20%3D%20nx%5E%7Bn-1%7D)
![(2) \frac{d}{dx}(f(x) .g(x))= f'(x)g(x)+f(x)g'(x)](https://tex.z-dn.net/?f=%282%29%20%5Cfrac%7Bd%7D%7Bdx%7D%28f%28x%29%20.g%28x%29%29%3D%20f%27%28x%29g%28x%29%2Bf%28x%29g%27%28x%29)
A(t) = t⁴(5t+4)
Differentiate with respect to t
![\frac{d}{dt} A(t)=\frac{d}{dt} [t^4(5t+4]](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdt%7D%20A%28t%29%3D%5Cfrac%7Bd%7D%7Bdt%7D%20%5Bt%5E4%285t%2B4%5D)
![\Rightarrow \frac{dA(t)}{dt} =(5t+4)\frac{dt^4}{dt} +t^4\frac{d}{dt} (5t+4)](https://tex.z-dn.net/?f=%5CRightarrow%20%5Cfrac%7BdA%28t%29%7D%7Bdt%7D%20%3D%285t%2B4%29%5Cfrac%7Bdt%5E4%7D%7Bdt%7D%20%2Bt%5E4%5Cfrac%7Bd%7D%7Bdt%7D%20%285t%2B4%29)
![\Rightarrow \frac{dA(t)}{dt} = (5t+4)4t^{4-1}+t^4.5](https://tex.z-dn.net/?f=%5CRightarrow%20%5Cfrac%7BdA%28t%29%7D%7Bdt%7D%20%3D%20%285t%2B4%294t%5E%7B4-1%7D%2Bt%5E4.5)
![\Rightarrow \frac{dA(t)}{dt} =4t^3(5t+4)+5t^4](https://tex.z-dn.net/?f=%5CRightarrow%20%5Cfrac%7BdA%28t%29%7D%7Bdt%7D%20%3D4t%5E3%285t%2B4%29%2B5t%5E4)
![\Rightarrow \frac{dA(t)}{dt} = 20t^4+16t^3+5t^4](https://tex.z-dn.net/?f=%5CRightarrow%20%5Cfrac%7BdA%28t%29%7D%7Bdt%7D%20%3D%2020t%5E4%2B16t%5E3%2B5t%5E4)
![\Rightarrow \frac{dA(t)}{dt} = 25t^4+16t^3](https://tex.z-dn.net/?f=%5CRightarrow%20%5Cfrac%7BdA%28t%29%7D%7Bdt%7D%20%3D%2025t%5E4%2B16t%5E3)
Therefore the rate of change of area
square inches/ s
What is the value of what?
Step-by-step explanation:
For #7 and #8, they have given you the answer of the product but without decimals.
You can just add up all the decimal places for this kind of multiplication.
For example, 6.<u>543</u> and 2.<u>6</u> have 4 decimal places in total (underlined), so just count four: 17.<u>0118</u>.
Correct they are
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