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Answer:
24. see below for a drawing with axis intercepts labeled
30. the plane is parallel to the y-z plane. It is "x=4". The only axis intercept is (4, 0, 0).
Step-by-step explanation:
24. The axis intercepts are fairly easily found. You can do it a couple of ways.
<u>first way</u>
- Set other variables to zero and divide by the coefficient of the variable of interest.
For 3x +5y +15z = 15, the x-intercept can be found by setting y and z to zero. This gives ...
3x = 15
so, dividing by 3 tells you the x-intercept is ...
x = 15/3 = 5
Likewise, the y- and z-intercepts are, respectively, ...
y = 15/5 = 3
z = 15/15 = 1
So, the points where the plane crosses the axes are ...
(5, 0, 0), (0, 3, 0), (0, 0, 1) . . . . axes intercepts
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<u>second way</u>
- Divide by the constant on the right, and express all coefficients as denominators
Doing that gives ...
The denominators are the corresponding intercepts. (This is called the "intercept form" of the equation of the plane.)
From the above equation, we see the axis intercepts are ...
(5, 0, 0), (0, 3, 0), (0, 0, 1) . . . . same as above
Actually, the math for these two methods is virtually identical. The second way does it "all at once", so can take fewer steps. Effectively, you're doing the math of the "first way" and expressing the result in the denominator.
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Please note that when a variable is missing, there is no intercept for that axis. That is, the plane is parallel to that axis. (In problem 30, for example, the y- and z-variables are missing. The plane x=4 is parallel to the y- and z-axes (the y-z plane).)
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In the first attached drawing, the x-axis is red, the y-axis is green, and the z-axis is blue. The labeling of the points is difficult to see, but the axes are a different (lighter) color behind the plane than in front of it. This plane forms a triangle in the first octant (x, y, z > 0).
In the second attached drawing, the x=0 plane is blue; the y=0 plane is green, and the z=0 plane is red. The plane of the equation is yellow. This figure better shows the triangle in the first octant.
3-D graph paper (isometric graph paper) grids are available with a web search. Such might make it easier to draw the planes you need to draw. (See the third attachment for one example.)
Answer:
the apprentice can complete the job in 30 hours by working alone.
Step-by-step explanation:
Given:
Number of hours required to complete a job by welder = 18 hours
Number of hours welder work on job = 6 hours.
Number of hours required by apprentice to complete the job = 14 hours
We need to find Number of hours required to complete a job by apprentice alone.
Solution:
Let Number of hours required to complete a job by apprentice alone be 'a'.
Also let the job completed be = 1
Now we know that ;
Time required on job is equal to sum of Number of hours welder work on job and Number of hours required by apprentice to complete the job.
framing in equation form we get
Time required on job =
Now we can say that;
each has done a fraction of the work so we will add to two fraction as number of hours of work done by Total number of hours required to do the work to complete 1 job.
so we can frame the equation as;
By reducing the fraction we get;
Now we will make the denominator common to solve the fraction we get;
Now denominators are same so we will solve the numerator we get;
Multiplying both side by 3a we get;
Combining the like terms we get;
Dividing both side by 2 we get;
Hence the apprentice can complete the job in 30 hours by working alone.