A sum of money at compound interest amounts to fifth-hold itself in 7 years. In how many years will it be 25 times itself?
To find the values of \(\sin x\) given the equation \(8\sin x = 4 + \cos x\), we can use trigonometric identities to rewrite the equation in terms of a single trigonometric function. The given equation is: \[8\sin x = 4 + \cos x\] We know the Pythagorean identity: \(\sin^2 x + \cos^2 x = 1\). Our goRead more
To find the values of \(\sin x\) given the equation \(8\sin x = 4 + \cos x\), we can use trigonometric identities to rewrite the equation in terms of a single trigonometric function.
The given equation is:
\[8\sin x = 4 + \cos x\]
We know the Pythagorean identity: \(\sin^2 x + \cos^2 x = 1\). Our goal is to express \(\cos x\) in terms of \(\sin x\) (or vice versa) to solve for \(\sin x\). Since the equation involves both \(\sin x\) and \(\cos x\), and we’re looking to find \(\sin x\), let’s isolate \(\cos x\) and then use the Pythagorean identity.
Rearrange the given equation to isolate \(\cos x\):
\[\cos x = 8\sin x – 4\]
Using the Pythagorean identity \(\cos^2 x = 1 – \sin^2 x\), we substitute for \(\cos x\) in the equation:
\[1 – \sin^2 x = (8\sin x – 4)^2\]
Expanding the right side and moving all terms to one side gives us a quadratic equation in \(\sin x\):
\[1 – \sin^2 x = 64\sin^2 x – 64\sin x + 16\]
Combine like terms:
\[65\sin^2 x – 64\sin x + 15 = 0\]
This is a quadratic equation in \(\sin x\). To solve for \(\sin x\), we use the quadratic formula where \(a = 65\), \(b = -64\), and \(c = 15\):
\[\sin x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}\]
\[= \frac{64 \pm \sqrt{(-64)^2 – 4 \cdot 65 \cdot 15}}{2 \cdot 65}\]
\[= \frac{64 \pm \sqrt{4096 – 3900}}{130}\]
\[= \frac{64 \pm \sqrt{196}}{130}\]
\[= \frac{64 \pm 14}{130}\]
So, we have two possible solutions for \(\sin x\):
1. \(\sin x = \frac{64 + 14}{130} = \frac{78}{130} = \frac{39}{65} = \frac{3}{5}\)
2. \(\sin x = \frac{64 – 14}{130} = \frac{50}{130} = \frac{25}{65} = \frac{5}{13}\)
Therefore, the values of \(\sin x\) that satisfy the given equation are \(\frac{3}{5}\) and \(\frac{5}{13}\).
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Let's denote the principal amount as \(P\), the rate of compound interest as \(r\), and the number of years as \(n\). According to the given information, the amount becomes fivefold in 7 years. Therefore, we can write the compound interest formula as follows: \[ A = P(1 + r)^n \] For the amount to bRead more
Let’s denote the principal amount as \(P\), the rate of compound interest as \(r\), and the number of years as \(n\).
According to the given information, the amount becomes fivefold in 7 years. Therefore, we can write the compound interest formula as follows:
\[ A = P(1 + r)^n \]
For the amount to be fivefold:
\[ 5P = P(1 + r)^7 \]
Simplifying:
\[ (1 + r)^7 = 5 \]
Now, we need to find in how many years the amount will be 25 times itself. We can set up a similar equation:
\[ 25P = P(1 + r)^n \]
\[ (1 + r)^n = 25 \]
We know that \((1 + r)^7 = 5\), so we can express 25 in terms of 5:
\[ (1 + r)^n = 5^2 \]
\[ (1 + r)^n = ((1 + r)^7)^2 \]
\[ (1 + r)^n = (1 + r)^{14} \]
Thus, we can see that \(n = 14\).
Therefore, it will take 14 years for the sum of money to become 25 times itself at the same rate of compound interest.
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