-0.030
Fit Straight Line to Data
The problem involves fitting a straight line to a data set where the variables are $(\ln x, y)$. The equation of a straight line is generally given by $Y = mX + c$. In this case, $X = \ln x$ and $Y = y$, so the equation becomes:
$y = m(\ln x) + c$
We are given the slope $m = -0.02$. The line intercepts the abscissa (the $\ln x$-axis) at $\ln x = 0.1$. This means that when $y=0$, $\ln x = 0.1$. We can use this point $(0.1, 0)$ to find the y-intercept $c$.
Calculate Line Intercept
Substitute the known point and slope into the line equation:
$0 = (-0.02)(0.1) + c$
$0 = -0.002 + c$
$c = 0.002$
Therefore, the equation of the fitted line is:
$y = -0.02(\ln x) + 0.002$
Calculate Value at x = 5
We need to find the value of $y$ when $x = 5$. First, calculate $\ln 5$.
$\ln 5 \approx 1.6094$
Now substitute $x=5$ (and thus $\ln x \approx 1.6094$) into the line equation:
$y = -0.02(\ln 5) + 0.002$
$y \approx -0.02(1.6094) + 0.002$
$y \approx -0.032188 + 0.002$
$y \approx -0.030188$
Rounding to three decimal places, the value of $y$ at $x=5$ is approximately $-0.030$.
Important Questions from Numerical Estimation
The number of significant figures in $\frac{25 \times 0.524}{100.0} = 0.131$ is
1200 men and 500 women can build a bridge in 2 weeks. 900 men and 250 women will take 3 weeks to build the same bridge. How many men will be needed to build the bridge in one week?
The number of 3-digit numbers such that the digit 1 is never to the immediate right of 2 is
Students applying for hostel rooms are allotted rooms in order of seniority. Students already staying in a room will move if they get a room in their preferred list. Preferences of lower ranked applicants are ignored during allocation.
Given the data below, which room will Ajit stay in?
Names
Student seniority
Current
Room preference list
Amar
1
P
R, S, Q
Akbar
2
None
R, S
Anthony
3
Q
P
Ajit
4
S
Q, P, R
There are 16 teachers who can teach Thermodynamics (TD), 11 who can teach Electrical Sciences (ES), and 5 who can teach both TD and Engineering Mechanics (EM). There are a total of 40 teachers. 6 cannot teach any of the three subjects, i.e. EM, ES or TD. 6 can teach only ES. 4 can teach all three subjects, i.e. EM, ES and TD. 4 can teach ES and TD. How many can teach both ES and EM but not TD?
