If a = 17.0079, b = 2.05, and c = 3.1415926, then what is a/b + c?
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It would take 10.7 years.
The formula for continuously compounded interest is:
where P is the principal, r is the interest rate as a decimal number, and t is the number of years.
Using our information we have:
We want to know when it will double the principal; therefore we substitute 2P for A and solve for t:
Divide both sides by P:
Take the natural log, ln, of each side to "undo" e:
Divide both sides by 0.065:
The formula for continuously compounded interest is:
where P is the principal, r is the interest rate as a decimal number, and t is the number of years.
Using our information we have:
We want to know when it will double the principal; therefore we substitute 2P for A and solve for t:
Divide both sides by P:
Take the natural log, ln, of each side to "undo" e:
Divide both sides by 0.065:
To give you a context on the problem, a tangent line is a line that intersects the parabola only at one single point. A parabola is a curve that forms an arc-shaped figure. A tangent line to a parabola is shown in the attached picture.
Now, we apply the concepts in calculus and analytical geometry. The first derivative of the equation is equal to the slope at the point of intersection. This slope must be equal to the slope of the tangent line.
y = x² - 5x + 7
dy/dx = slope = 2x -5
Since tangent lines must have the same slope with what they intersect with, we can determine the slope from the equation: y = 3x + c. This is already arranged in a slope-intercept form, where 3 is the slope and c is the y-intercept. So, we can equate the equation above to 3.
2x - 5 = 3
x = 4
Now, we substitute x=4 to the original equation of the parabola:
y = (4)² - 5(4) + 7
y = 3
Therefore, the point of intersection is at (4,3). Now, we use it to the equation of the tangent line to find c.
y = 3x + c
3 = 3(4) + c
c = -9
Now, we apply the concepts in calculus and analytical geometry. The first derivative of the equation is equal to the slope at the point of intersection. This slope must be equal to the slope of the tangent line.
y = x² - 5x + 7
dy/dx = slope = 2x -5
Since tangent lines must have the same slope with what they intersect with, we can determine the slope from the equation: y = 3x + c. This is already arranged in a slope-intercept form, where 3 is the slope and c is the y-intercept. So, we can equate the equation above to 3.
2x - 5 = 3
x = 4
Now, we substitute x=4 to the original equation of the parabola:
y = (4)² - 5(4) + 7
y = 3
Therefore, the point of intersection is at (4,3). Now, we use it to the equation of the tangent line to find c.
y = 3x + c
3 = 3(4) + c
c = -9