1. Determine whether the lines given by the equations below are parallel, perpendicular, or neither. Also, find a rigorous algebraic solution for each problem.
To determine whether the given lines are parallel, perpendicular, or neither, we can compare their slopes. The slope-intercept form of a linear equation is y = mx + b, where m is the slope.
Let’s analyze each problem one by one:
a. {3y + 4x = 12, -6y = 8x + 1}
To find the slope of the first equation, we need to rearrange it into slope-intercept form:
3y + 4x = 12
3y = -4x + 12
y = (-4/3)x + 4
The slope of the first equation is -4/3.
For the second equation, we need to rearrange it as well:
-6y = 8x + 1
y = (-8/6)x – 1/6
y = (-4/3)x – 1/6
The slope of the second equation is also -4/3.
Since the slopes of both equations are the same (-4/3), the lines are parallel.
b. {3y + x = 12, -y = 8x + 1}
To find the slope of the first equation:
3y + x = 12
3y = -x + 12
y = (-1/3)x + 4
The slope of the first equation is -1/3.
For the second equation:
-y = 8x + 1
y = -8x – 1
The slope of the second equation is -8.
Since the slopes are different (-1/3 and -8), the lines are neither parallel nor perpendicular.
c. {4x – 7y = 10, 7x + 4y = 1}
To find the slope of the first equation:
4x – 7y = 10
-7y = -4x + 10
y = (4/7)x – 10/7
The slope of the first equation is 4/7.
For the second equation:
7x + 4y = 1
4y = -7x + 1
y = (-7/4)x + 1/4
The slope of the second equation is -7/4.
Since the slopes are different (4/7 and -7/4), the lines are neither parallel nor perpendicular.
To summarize:
a. The lines in problem a) are parallel.
b. The lines in problem b) are neither parallel nor perpendicular.
c. The lines in problem c) are neither parallel nor perpendicular.
2. A ball is thrown in the air from the top of a building. Its height, in meters above ground, as a function of time, in seconds, is given by h(t)=-4.9t^2+24t+8 . What is the height of the building? What is the maximum height reached by the ball? How long does it take to reach maximum height? Also, find a rigorous algebraic solution for the problem.
To find the height of the building, we need to determine the value of h(t) when the ball is at the ground level. In this case, the height of the building corresponds to the value of h(t) when t = 0.
Let’s substitute t = 0 into the equation h(t) = -4.9t^2 + 24t + 8:
h(0) = -4.9(0)^2 + 24(0) + 8
= 0 + 0 + 8
= 8
Therefore, the height of the building is 8 meters.
To find the maximum height reached by the ball, we need to determine the vertex of the parabolic function h(t) = -4.9t^2 + 24t + 8. The vertex represents the highest point on the graph and corresponds to the maximum height.
The formula for the x-coordinate of the vertex of a parabola in the form y = ax^2 + bx + c is given by x = -b / (2a). In this case, a = -4.9 and b = 24.
x = -24 / (2(-4.9))
= -24 / (-9.8)
= 2.449
Now, let’s substitute this value of x back into the equation to find the maximum height:
h(2.449) = -4.9(2.449)^2 + 24(2.449) + 8
= -4.9(5.999401) + 58.776 + 8
= -29.396 + 58.776 + 8
= 37.38
Therefore, the maximum height reached by the ball is approximately 37.38 meters.
To find the time it takes to reach the maximum height, we can use the x-coordinate of the vertex, which we found to be 2.449 seconds.
Therefore, it takes approximately 2.449 seconds for the ball to reach its maximum height.
To find a rigorous algebraic solution for the problem, we can use the formula for the vertex of a parabola and the quadratic formula.
The vertex formula for a parabola in the form y = ax^2 + bx + c is given by x = -b / (2a). In this case, a = -4.9 and b = 24.
x = -24 / (2(-4.9))
= -24 / (-9.8)
= 2.449
We have found the x-coordinate of the vertex, which is the time it takes for the ball to reach its maximum height.
To find the maximum height, substitute this value of x back into the equation:
h(2.449) = -4.9(2.449)^2 + 24(2.449) + 8
= -4.9(5.999401) + 58.776 + 8
= -29.396 + 58.776 + 8
= 37.38
Therefore, the maximum height reached by the ball is approximately 37.38 meters.
To summarize:
– The height of the building is 8 meters.
– The maximum height reached by the ball is approximately 37.38 meters.
– It takes approximately 2.449 seconds for the ball to reach its maximum height.