The point (1, 1) satisfies the given system of equations y = 3x − 2 and y = -2x + 3.
Given the system of equations:y = 3x - 2 (Equation 1)y = -2x + 3 (Equation 2)To find the point which satisfies the given system of equations, we have to equate Equation 1 and Equation 2.y = 3x - 2 = -2x + 3y + 2x = 3 + 2x = 5x = 5/5 = 1Putting the value of x in any of the equation, we can find the value of y.y = 3(1) - 2y = 3 - 2 = 1
The graph of the given system of equations:y = 3x - 2 and y = -2x + 3 is shown below:Graph of the given system of equations.
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Debt (Kd) 8.5 % 20 % 1.70 %
Preferred stock (Kp) 7.2 10 0.72
Common equity (Ke) (retained earnings) 7.5 70 5.25
Weighted average cost of capital (Ka) 7.67 %
A) If the firm has $42 million in retained earnings, at what size capital structure will the firm run out of retained earnings? (Enter your answer in millions of dollars (e.g., $10 million should be entered as "10").)
B) The 8.5 percent cost of debt referred to earlier applies only to the first $14 million of debt. After that the cost of debt will go up. At what size capital structure will there be a change in the cost of debt? (Enter your answer in millions of dollars (e.g., $10 million should be entered as "10").)
A) , the firm will run out of retained earnings at a capital structure size of $560 million for the first capital structure, $60 million for the second capital structure, and $800 million for the third capital structure.
B) there will be a change in the cost of debt at a capital structure size of approximately $19.95 million.
A) To determine the size of the capital structure at which the firm will run out of retained earnings, we need to calculate the retained earnings limit. Retained earnings are expressed as a percentage of the common equity (Ke). From the given information, the retained earnings are 7.5%, 70, and 5.25 for the three different capital structures. Let's assume the size of the capital structure as "x" million dollars.
For the first capital structure (7.5% Ke), the retained earnings limit is given by:
0.075x = 42
x = 42 / 0.075
x ≈ 560
For the second capital structure (70 Ke), the retained earnings limit is given by:
0.70x = 42
x = 42 / 0.70
x ≈ 60
For the third capital structure (5.25 Ke), the retained earnings limit is given by:
0.0525x = 42
x = 42 / 0.0525
x ≈ 800
Therefore, the firm will run out of retained earnings at a capital structure size of $560 million for the first capital structure, $60 million for the second capital structure, and $800 million for the third capital structure.
B) The 8.5% cost of debt applies only to the first $14 million of debt. After that, the cost of debt will go up. To find the capital structure size at which the cost of debt changes, we need to consider the different debt levels and their associated costs.
Assuming the capital structure size as "y" million dollars, we can calculate the change in cost of debt:
For the first $14 million of debt, the cost of debt is 8.5%. Therefore, we have:
0.085(14) = 1.19
Once the debt exceeds $14 million, the cost of debt will change. From the given information, the cost of debt increases to 20% after $14 million. So, we have:
0.20(y - 14) = 0.20y - 2.8
To find the size of the capital structure at which the cost of debt changes, we need to equate the two expressions:
1.19 = 0.20y - 2.8
0.20y = 1.19 + 2.8
0.20y = 3.99
y = 3.99 / 0.20
y ≈ 19.95
Therefore, there will be a change in the cost of debt at a capital structure size of approximately $19.95 million.
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8. Tara pays a base rate for her long distance phone service plus a per-minute charge.
The graph below shows what she would pay for her long distance phone service for the
first 60 minutes. What does the y-intercept of this graph represent?
9. Rich is a number of a gym. He pays a monthly fee plus a per-visit fee. The equation
The y-intercept of the graph, of b = 5, represents that Tara pays a base fee of $5.
How to define a linear function?The slope-intercept equation for a linear function is presented as follows:
y = mx + b
In which:
m is the slope.b is the intercept.The intercept on the graph is the value of y at which the graph touches or crosses the y-axis, hence it is given as follows:
b = 5.
At the y-intercept, we have that x = 0, hence it represents the base fee of Tara's plan.
Missing InformationQuestion 9 is incomplete, hence it cannot be answered.
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The product of sin 30° and sin 60°
The multiplication of sin 30° and sin 60° results in √3/2, representing the value of their product without any alteration.
To find the product of sin 30° and sin 60°, we can use the trigonometric identity for the sine of the sum of two angles. The identity states that sin(A + B) = sin(A)cos(B) + cos(A)sin(B).
Let's consider sin 30° and sin 60°:
sin 30° = 1/2
sin 60° = √3/2
Using the identity, we have:
sin (30° + 60°) = sin 30°cos 60° + cos 30°sin 60°
Now, we can substitute the values:
sin (30° + 60°) = (1/2)(√3/2) + (√3/2)(1/2)
Simplifying, we get:
sin (30° + 60°) = √3/4 + √3/4
Combining like terms, we have:
sin (30° + 60°) = 2√3/4
Now, let's simplify the expression:
sin (30° + 60°) = √3/2
Therefore, the product of sin 30° and sin 60° is √3/2
In summary, the product of sin 30° and sin 60° is equal to √3/2, which can be derived by using the trigonometric identity for the sine of the sum of two angles.
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