Question
Find a value for k so that the line through (35, k) and (42, 27) is parallel to the line with equation y = x.
The correct answer is: Value of k for the given line is 20
Hint:-
1. Slopes of parallel lines are equal.
2. When we have 2 points that lie on a given line then we can find the equation of the said line by using the 2-point formula-
(y-y1) = (y2-y1) * (x-x1)
(x2-x1)
Step-by-step solution:-
y = x
∴ y = x + 0.
Comparing the above equation with standard form of a line i.e. y = mx + c, we get- m = 1 ......................... (Equation i)
The given line is parallel to y = x
and we know that parallel lines have equal slopes.
∴ Slope of the given line = slope of line (y = x)
∴ Slope of the given line = m = 1 ..................... (From Equation i) .................................................................... (Equation ii)
The given line passes through the point (35,k) & (42,27).
Hence, x1 = 35, y1 = k, x2 = 42 & y2 = 27
(y-y1) = (y2-y1) * (x-x1)
(x2-x1)
∴ (y-k) = (27-k) * (x- 35)
42-35
∴ y - k = 27 - k * (x - 35)
7
∴ 7 * (y - k) = (27 - k) * (x - 35) ............................................................................ (Multiplying both sides by 7)
∴ 7y - 7k = 27x - 945 - kx + 35k
∴ 7y = 27x - kx + 35k + 7k - 945
∴ 7y = (27 - k) x + 42k - 945
∴ y = (27 - k)/7 x + 6k - 135 ........................... (Dividing both sides by 7) ........................................... (Equation iii)
Comparing Equation iii with the standard form of a straight line i.e. y = mx + c, we get-
m = (27 - k) / 7
∴ 1 = (27 - k) / 7 ..................................... (From Equation ii)
∴ 7 = 27 - k ...................................... (Multiplying both sides by 7)
∴ k = 27 - 7
∴ k = 20.
Final Answer:-
∴ Value of k for the given line is 20.
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Carrie, a packaging engineer, is designing a container to hold 12 drinking glasses shaped as regular octagonal prisms. Her initial sketch of the top view of the base of the container is shown above.
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Note:
There is a shorter way of doing this problem. There is a simple rule followed while increasing or decreasing a value by a certain percentage. When we decrease a quantity by x%, we multiply the quantity by (1-0.x). Whereas when we increase a quantity by x%, we multiply it with (1+0.x).
Carrie, a packaging engineer, is designing a container to hold 12 drinking glasses shaped as regular octagonal prisms. Her initial sketch of the top view of the base of the container is shown above.
Carrie redesigned the container because the initial sketch did not account for cushioning material between the glasses. The area of the base of the newly designed container is greater than the area of the base in the initial sketch. What is the area, in square inches, of the base of the newly designed container?
Note:
There is a shorter way of doing this problem. There is a simple rule followed while increasing or decreasing a value by a certain percentage. When we decrease a quantity by x%, we multiply the quantity by (1-0.x). Whereas when we increase a quantity by x%, we multiply it with (1+0.x).
Find the equation of line that passes through the point (2, -9) and which is perpendicular to the line x = 5.
Use the perpendicular line formula to determine whether two given lines are perpendicular. For example, when the slope of two lines is given to compare, we can use the perpendicular line's formula. A 90-degree angle is created by two lines that are perpendicular to one another.
Slope exists on every line. Because it shows how quickly our line is rising or falling, the slope of a line reveals how steep a line is. Mathematically, the slope of a line is known as the ratio of change in the line's y-value to the change in its x-value.
¶A line's slope can be determined using its two points (x1, y1) and (x2, y2). The formula (y2 - y1) / is used to find the change in y and divided by the change in x. (x2 - x1).
Find the equation of line that passes through the point (2, -9) and which is perpendicular to the line x = 5.
Use the perpendicular line formula to determine whether two given lines are perpendicular. For example, when the slope of two lines is given to compare, we can use the perpendicular line's formula. A 90-degree angle is created by two lines that are perpendicular to one another.
Slope exists on every line. Because it shows how quickly our line is rising or falling, the slope of a line reveals how steep a line is. Mathematically, the slope of a line is known as the ratio of change in the line's y-value to the change in its x-value.
¶A line's slope can be determined using its two points (x1, y1) and (x2, y2). The formula (y2 - y1) / is used to find the change in y and divided by the change in x. (x2 - x1).
Write the equations of the given lines.
Line 1: y-intercept = 3, slope = 2
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Write the equations of the given lines.
Line 1: y-intercept = 3, slope = 2
Line 2: y-intercept = - 1, slope = -5
Carrie, a packaging engineer, is designing a container to hold 12 drinking glasses shaped as regular octagonal prisms. Her initial sketch of the top view of the base of the container is shown above.
If the length and width of the container base in the initial sketch were doubled, at most how many more glasses could the new container hold?
Note:
A few simple ideas are used in solving this problem, like, area of a rectangle is given by the product of its length and breadth and the basic idea of division.
Carrie, a packaging engineer, is designing a container to hold 12 drinking glasses shaped as regular octagonal prisms. Her initial sketch of the top view of the base of the container is shown above.
If the length and width of the container base in the initial sketch were doubled, at most how many more glasses could the new container hold?
Note:
A few simple ideas are used in solving this problem, like, area of a rectangle is given by the product of its length and breadth and the basic idea of division.
