Two integers have a sum of 11. The sum of the greater integer squared and 30 times the smaller integer is 205. What are the two integers?
x+y = 11
30x+y^2 = 205
eliminate y to get
30x + (11-x)^2 = 205
now solve as usual
The sum of the greater integer squared and 30 times the smaller integer is 205. What are the two integers and how to solve
Let the two integers be x and y, where x is the greater integer and y is the smaller integer.
From the problem, we have two equations:
x + y = 11 ---(1)
x^2 + 30y = 205 ---(2)
We want to solve for x and y.
From equation (1), we have:
x = 11 - y
Substituting this into equation (2), we get:
(11 - y)^2 + 30y = 205
Expanding and simplifying, we get:
y^2 - 22y + 36 = 0
This is a quadratic equation that we can solve using the quadratic formula:
y = (-(-22) ± sqrt((-22)^2 - 4(1)(36))) / (2(1))
y = (22 ± sqrt(244)) / 2
y = 11 ± sqrt(61)
Since y is the smaller integer, we take the negative square root:
y = 11 - sqrt(61)
Substituting this into equation (1), we get:
x + 11 - sqrt(61) = 11
x = sqrt(61)
So the two integers are:
x = sqrt(61) (the greater integer)
y = 11 - sqrt(61) (the smaller integer)
Gary is a fitness trainer that makes 30 and 45 minute appointments with clients. Gary’s contract allows him to make up to 30 hours of appointments in a week. State the inequality that represents all possible combinations of appointments that Gary can make in a week.
Let x be the number of 30-minute appointments that Gary makes in a week, and y be the number of 45-minute appointments that he makes in a week.
One 30-minute appointment takes 0.5 hours (30/60), and one 45-minute appointment takes 0.75 hours (45/60).
Thus, the total number of hours of appointments that Gary makes in a week is:
0.5x + 0.75y
According to Gary's contract, he can make up to 30 hours of appointments in a week. Therefore, we can write the following inequality:
0.5x + 0.75y ≤ 30
This is the inequality that represents all possible combinations of appointments that Gary can make in a week, given the constraints of his contract.
Gary is a fitness trainer that makes 30 and 45 minute appointments with clients. Gary’s contract allows him to make up to 30 hours of appointments in a week. State the inequality that represents all possible combinations of appointments that Gary can make in a week. Graph the inequality.
The inequality that represents all possible combinations of appointments that Gary can make in a week is:
0.5x + 0.75y ≤ 30
To graph this inequality, we first need to rewrite it in slope-intercept form:
0.75y ≤ -0.5x + 30
y ≤ (-0.5/0.75)x + 40
The inequality is a linear equation in two variables, x and y. To plot the graph, we can first plot the boundary line:
y = (-0.5/0.75)x + 40
This line has a y-intercept of 40 and a slope of -0.5/0.75 = -2/3.
We can plot the y-intercept first, and then use the slope to find additional points on the line.
Next, we need to determine which side of the line represents the solution to the inequality. Because the inequality has a "≤" symbol, we want to shade the region below the line.
Here is the graph of the inequality:
Consider the inequality y he’s less then or equal to ax+6. The point. (2,1) is in the solution region when is what
We can substitute the given point into the inequality and solve for the value of a.
y ≤ ax + 6
Substituting (2,1):
1 ≤ a(2) + 6
1 ≤ 2a + 6
Subtracting 6 from both sides:
-5 ≤ 2a
Dividing both sides by 2:
-2.5 ≤ a
Therefore, the point (2,1) is in the solution region when a is greater than or equal to -2.5.
In interval notation, the solution is:
a ∈ [-2.5, ∞)