Determine the equation of a line that is perpendicular to the line 2x+3y=7 and passes through the point (0,6)
What is the volume if the prism in terms of y?
1st dimension: 2y+1
2nd dimension: y
3rd dimension: y+3
we need slope of 3/2, so the line is
y-6 = 3/2 x
prism: volume = (2y+1)(y)(y+3)
To determine the equation of a line that is perpendicular to the line 2x+3y=7, we need to find the slope of the given line and then find the negative reciprocal to get the slope of the perpendicular line.
Step 1: Convert the given equation into slope-intercept form (y=mx+b), where m is the slope.
2x + 3y = 7
3y = -2x + 7
y = (-2/3)x + 7/3
Step 2: Identify the slope (m) of the given line.
From the slope-intercept form, we see that the slope of the given line is -2/3.
Step 3: Find the negative reciprocal of the slope.
To obtain the slope of the line perpendicular to the given line, we flip the fraction and change the sign.
The negative reciprocal of -2/3 is 3/2.
Step 4: Use the point-slope form (y - y1 = m(x - x1)) to find the equation of the line perpendicular to the given line that passes through the point (0,6).
Let's use the point (0,6) and the slope 3/2 in the point-slope form.
y - 6 = (3/2)(x - 0)
y - 6 = (3/2)x
y = (3/2)x + 6
Therefore, the equation of the line that is perpendicular to the line 2x+3y=7 and passes through the point (0,6) is y = (3/2)x + 6.
Now, onto the second question:
To calculate the volume of a prism in terms of y, we need to multiply all three dimensions together. Given that the dimensions of the prism are:
1st dimension: 2y + 1
2nd dimension: y
3rd dimension: y + 3
The volume (V) is given by:
V = (2y + 1) * y * (y + 3)
To find the final volume, you can simplify the expression by multiplying the terms together and combining like terms.