The lengths of the sides of a rectangular window have the ratio 1.5 to 1. The area of the window is 1,350 square inches. What are the dimensions of the window?
Let's call the length of the shorter side of the window "x". Then the length of the longer side of the window is 1.5x, since the ratio of the lengths is 1.5 to 1.
The area of a rectangle is found by multiplying the length by the width. So we can set up the equation:
x * (1.5x) = 1350
Simplifying the equation, we have:
1.5x^2 = 1350
Dividing both sides of the equation by 1.5, we get:
x^2 = 900
Taking the square root of both sides, we find the value of x:
x = √900
x = 30
So the length of the shorter side of the window is 30 inches, and the length of the longer side is 1.5 * x = 1.5 * 30 = 45 inches.
Therefore, the dimensions of the window are 30 inches by 45 inches.
What is the solution to the following equation?
x^2+14x+48=0
To solve the quadratic equation x^2 + 14x + 48 = 0, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a),
where a, b, and c are the coefficients of the equation.
In this case, a = 1, b = 14, and c = 48.
Substituting these values into the quadratic formula, we have:
x = (-(14) ± √((14)^2 - 4(1)(48))) / (2(1))
Simplifying the expression under the square root, we get:
x = (-14 ± √(196 - 192)) / 2
x = (-14 ± √4) / 2
x = (-14 ± 2) / 2
This gives us two possible answers for x:
x = (-14 + 2) / 2 = -12 / 2 = -6
x = (-14 - 2) / 2 = -16 / 2 = -8
Hence, the solutions to the equation x^2 + 14x + 48 = 0 are x = -6 and x = -8.
Max makes and sells posters. The function p(x) = –10x^2 + 200x – 250, graphed below, indicates how much profit he makes in a month if he sells the posters for 20 – x dollars each.
The graph of a downward-facing parabola is shown in the first quadrant of a coordinate grid. The parabola connects the points left-parenthesis 2 comma 100 right-parenthesis, left-parenthesis 10 comma 750 right-parenthesis, and left-parenthesis 18 comma 100 right-parenthesis.
What should Max charge per poster to make the maximum profit, and what is the maximum profit he can make in a month?
To find the price per poster that Max should charge to make the maximum profit, we need to determine the vertex of the parabola.
The vertex of a parabola in the form y = ax^2 + bx + c is given by the values:
x = -b / (2a),
y = f(x) = c - (b^2 / (4a)).
In our case, the quadratic function is p(x) = -10x^2 + 200x - 250. Comparing with the general form, we can see that a = -10, b = 200, and c = -250.
Using the formulas, we have:
x = -(200) / (2(-10)) = -200 / -20 = 10,
y = -250 - (200^2 / (4(-10))) = -250 - (40000 / (-40)) = -250 - (-1000) = 750.
Therefore, the vertex of the parabola is (10, 750). This means that Max should charge $10 per poster to maximize his profit.
To find the maximum profit he can make in a month, we substitute the x-value of the vertex (10) into the profit function p(x):
p(10) = -10(10)^2 + 200(10) - 250 = -10(100) + 2000 - 250 = -1000 + 2000 - 250 = 750.
So, the maximum profit Max can make in a month is $750.
Use graphing to find the solutions to the system of equations.
enlarged left-brace Start Layout 1st row x squared minus y equals 4 2nd row 2 x plus y equals negative 1 End Layout
To find the solutions to the system of equations visually, we can graph both equations and determine the points of intersection.
First, let's graph the equation x^2 - y = 4.
We can rewrite this equation in terms of y:
y = x^2 - 4.
Creating a table of values, we can choose a few x-values, calculate the corresponding y-values, and plot the points on a graph:
x | y
-------
-2 | 0
-1 | 3
0 | -4
1 | -3
2 | 0
Plotting these points and connecting them with a smooth curve, we obtain a U-shaped graph.
Next, let's graph the equation 2x + y = -1.
We can rewrite this equation in terms of y:
y = -2x - 1.
Using the same table of values as before, we choose x-values and calculate y-values:
x | y
-------
-2 | -3
-1 | -1
0 | -1
1 | -3
2 | -5
Plotting these points and connecting them with a straight line, we can observe that the line has a negative slope.
After graphing both equations, we can see that the line and the curve intersect at two points: (-1, -3) and (2, 0).
Therefore, the solutions to the system of equations x^2 - y = 4 and 2x + y = -1 are (-1, -3) and (2, 0).