Answer:
Step-by-step explanation:
From the information given:
We are to find; an expression for the total no. of articles written since 1983
∴
The total no. of articles written since 1983 ![=\int \limts ^t_0 U(t) dt](https://tex.z-dn.net/?f=%3D%5Cint%20%5Climts%20%5Et_0%20U%28t%29%20dt)
![= \int ^t_0 \limits \dfrac{4.1 e^{0.6\ t}}{0.2+e^{0.6t}} \ dt](https://tex.z-dn.net/?f=%3D%20%5Cint%20%5Et_0%20%5Climits%20%5Cdfrac%7B4.1%20e%5E%7B0.6%5C%20t%7D%7D%7B0.2%2Be%5E%7B0.6t%7D%7D%20%5C%20%20dt)
![=\dfrac{4.1}{0.6}\bigg [ \mathtt{In} \bigg | 0.2 + e^{0.6 x} \bigg| \bigg]^t_0](https://tex.z-dn.net/?f=%3D%5Cdfrac%7B4.1%7D%7B0.6%7D%5Cbigg%20%5B%20%5Cmathtt%7BIn%7D%20%5Cbigg%20%7C%200.2%20%2B%20e%5E%7B0.6%20x%7D%20%5Cbigg%7C%20%5Cbigg%5D%5Et_0)
![=\dfrac{4.1}{0.6}\bigg [ \mathtt{In} \bigg | 0.2 + e^{0.6 x} \bigg| - \mathtt{In} \bigg|0.2+1\bigg|\bigg]](https://tex.z-dn.net/?f=%3D%5Cdfrac%7B4.1%7D%7B0.6%7D%5Cbigg%20%5B%20%5Cmathtt%7BIn%7D%20%5Cbigg%20%7C%200.2%20%2B%20e%5E%7B0.6%20x%7D%20%5Cbigg%7C%20-%20%5Cmathtt%7BIn%7D%20%5Cbigg%7C0.2%2B1%5Cbigg%7C%5Cbigg%5D)
![=\dfrac{4.1}{0.6}\bigg [ \mathtt{In} \bigg | \dfrac{0.2 + e^{0.6 x}}{1.2} \bigg|\bigg]](https://tex.z-dn.net/?f=%3D%5Cdfrac%7B4.1%7D%7B0.6%7D%5Cbigg%20%5B%20%5Cmathtt%7BIn%7D%20%5Cbigg%20%7C%20%5Cdfrac%7B0.2%20%2B%20e%5E%7B0.6%20x%7D%7D%7B1.2%7D%20%5Cbigg%7C%5Cbigg%5D)
![=\dfrac{41}{6} \mathtt{In} \bigg | {0.2 + e^{0.6 x} \bigg|-\dfrac{41}{6} \mathtt{ In} \bigg | 1.2\bigg |](https://tex.z-dn.net/?f=%3D%5Cdfrac%7B41%7D%7B6%7D%20%5Cmathtt%7BIn%7D%20%5Cbigg%20%7C%20%7B0.2%20%2B%20e%5E%7B0.6%20x%7D%20%5Cbigg%7C-%5Cdfrac%7B41%7D%7B6%7D%20%5Cmathtt%7B%20In%7D%20%5Cbigg%20%7C%201.2%5Cbigg%20%7C)
![=6.833\times \mathtt{In} \bigg | {0.2 + e^{0.6 x} \bigg|-1.2458](https://tex.z-dn.net/?f=%3D6.833%5Ctimes%20%5Cmathtt%7BIn%7D%20%5Cbigg%20%7C%20%7B0.2%20%2B%20e%5E%7B0.6%20x%7D%20%5Cbigg%7C-1.2458)
![=6.833\times \mathtt{In} \bigg | {0.2 + e^{0.6 x} \bigg|-1.25 \ \mathbf{thousand \ articles}](https://tex.z-dn.net/?f=%3D6.833%5Ctimes%20%5Cmathtt%7BIn%7D%20%5Cbigg%20%7C%20%7B0.2%20%2B%20e%5E%7B0.6%20x%7D%20%5Cbigg%7C-1.25%20%5C%20%5Cmathbf%7Bthousand%20%5C%20articles%7D)
Therefore, the total number of articles written since 1983 is ![=6.833\times \mathtt{In} \bigg | {0.2 + e^{0.6 x} \bigg|-1.25 \ \mathbf{thousand \ articles}](https://tex.z-dn.net/?f=%3D6.833%5Ctimes%20%5Cmathtt%7BIn%7D%20%5Cbigg%20%7C%20%7B0.2%20%2B%20e%5E%7B0.6%20x%7D%20%5Cbigg%7C-1.25%20%5C%20%5Cmathbf%7Bthousand%20%5C%20articles%7D)
b. To find how many articles were being written from 1983 to 2003
i.e. t = 2003 - 1983 = 20
∴
Total articles written from 1983 to 2003 is ![=\int \limts ^{20}_0 U(t) dt](https://tex.z-dn.net/?f=%3D%5Cint%20%5Climts%20%5E%7B20%7D_0%20U%28t%29%20dt)
![= \int ^{20}_0 \limits \dfrac{4.1 e^{0.6\ t}}{0.2+e^{0.6t}} \ dt](https://tex.z-dn.net/?f=%3D%20%5Cint%20%5E%7B20%7D_0%20%5Climits%20%5Cdfrac%7B4.1%20e%5E%7B0.6%5C%20t%7D%7D%7B0.2%2Be%5E%7B0.6t%7D%7D%20%5C%20%20dt)
![=\dfrac{4.1}{0.6}\bigg [ \mathtt{In} \bigg | 0.2 + e^{0.6 x} \bigg| \bigg]^{20}_0](https://tex.z-dn.net/?f=%3D%5Cdfrac%7B4.1%7D%7B0.6%7D%5Cbigg%20%5B%20%5Cmathtt%7BIn%7D%20%5Cbigg%20%7C%200.2%20%2B%20e%5E%7B0.6%20x%7D%20%5Cbigg%7C%20%5Cbigg%5D%5E%7B20%7D_0)
![=\dfrac{4.1}{0.6}\bigg [ \mathtt{In} \bigg | 0.2 + e^{0.6 *20} \bigg| - \mathtt{In} \bigg|0.2+e^{0.6*0}\bigg|\bigg]](https://tex.z-dn.net/?f=%3D%5Cdfrac%7B4.1%7D%7B0.6%7D%5Cbigg%20%5B%20%5Cmathtt%7BIn%7D%20%5Cbigg%20%7C%200.2%20%2B%20e%5E%7B0.6%20%2A20%7D%20%5Cbigg%7C%20-%20%5Cmathtt%7BIn%7D%20%5Cbigg%7C0.2%2Be%5E%7B0.6%2A0%7D%5Cbigg%7C%5Cbigg%5D)
![=\dfrac{4.1}{0.6}\bigg [ \mathtt{In} \bigg | 0.2 + e^{12} \bigg| - \mathtt{In} \bigg|0.2+e^0\bigg|\bigg]](https://tex.z-dn.net/?f=%3D%5Cdfrac%7B4.1%7D%7B0.6%7D%5Cbigg%20%5B%20%5Cmathtt%7BIn%7D%20%5Cbigg%20%7C%200.2%20%2B%20e%5E%7B12%7D%20%5Cbigg%7C%20-%20%5Cmathtt%7BIn%7D%20%5Cbigg%7C0.2%2Be%5E0%5Cbigg%7C%5Cbigg%5D)
![=\dfrac{4.1}{0.6}\bigg [ \mathtt{In} \bigg | 0.2 + e^{12} \bigg| - \mathtt{In} \bigg|0.2+1\bigg|\bigg]](https://tex.z-dn.net/?f=%3D%5Cdfrac%7B4.1%7D%7B0.6%7D%5Cbigg%20%5B%20%5Cmathtt%7BIn%7D%20%5Cbigg%20%7C%200.2%20%2B%20e%5E%7B12%7D%20%5Cbigg%7C%20-%20%5Cmathtt%7BIn%7D%20%5Cbigg%7C0.2%2B1%5Cbigg%7C%5Cbigg%5D)
![=\dfrac{4.1}{0.6}\bigg [ \mathtt{In} \bigg | 0.2 + e^{12} \bigg| - \mathtt{In} \bigg|1.2\bigg|\bigg]](https://tex.z-dn.net/?f=%3D%5Cdfrac%7B4.1%7D%7B0.6%7D%5Cbigg%20%5B%20%5Cmathtt%7BIn%7D%20%5Cbigg%20%7C%200.2%20%2B%20e%5E%7B12%7D%20%5Cbigg%7C%20-%20%5Cmathtt%7BIn%7D%20%5Cbigg%7C1.2%5Cbigg%7C%5Cbigg%5D)
![=\dfrac{4.1}{0.6}\bigg [ \mathtt{In} \bigg | \dfrac{0.2 + e^{12}}{1.2} \bigg|\bigg]](https://tex.z-dn.net/?f=%3D%5Cdfrac%7B4.1%7D%7B0.6%7D%5Cbigg%20%5B%20%5Cmathtt%7BIn%7D%20%5Cbigg%20%7C%20%5Cdfrac%7B0.2%20%2B%20e%5E%7B12%7D%7D%7B1.2%7D%20%5Cbigg%7C%5Cbigg%5D)
= 80.75 thousand articles
Answer:
x = 10
The both line crosses with a little hyphen, meaning those are same
Answered by GAUTHMATH
Answer:
<h3>y + 2 = 3/7(x-7)</h3>
Step-by-step explanation:
The point-slope form of the equation will be expressed as;
y - y0 = m(x-x0) where;
m is the slope
(x0, y0) is the point on the line
Given
Slope m = 3/7
Point (x0, y0) = (7, -2)
Substitute into the equation;
y - y0 = m(x-x0)
y - (-2) = 3/7 (x - 7)
<em>y + 2 = 3/7(x-7)</em>
<em>Hence the equation in point-slope form is y + 2 = 3/7(x-7)</em>
Pls retake the pic it’s blurry
Answer:
The product of (4 +5x)(4-5x) is 16 - 25x².
Step-by-step explanation:
To find: Product of (4 +5x)(4-5x) using suitable identities.
Identity used : a² - b² = (a+b)(a-b)
Solution:
(4 +5x)(4-5x)
= (4)² - (5x)².........(using identity a² - b² = (a+b)(a-b))
(Here, a = 4 and b = 5x)
= 16 - 25x²
The given question can be solved without the use of the identity
a² - b² = (a+b)(a-b) as well.
Alternative Solution (without use of identity)-
(4 +5x)(4-5x)
(As (a+b)(c+d) = a(c+d)+b(c+d) )
= 4(4-5x) + 5x(4-5x)
= 16 - 20x + 20x - 25x²
= 16 - 25x²
Hence, the product of (4 +5x)(4-5x) is 16 - 25x².
#SPJ2
https://brainly.in/question/18367523