Math  /  Algebra

Question2. Solve for xx : 92x+1=(127)x19^{2 x+1}=\left(\frac{1}{27}\right)^{x-1} (4 marks)
3. An element has a half-life of 45 years. How long, in years, will it take for 8 g of the element to decay to 10 g ? ( 3 marks) A=A0(12)thA=A_{0}\left(\frac{1}{2}\right)^{\frac{t}{h}}

Caroline is looking to invest $5000\$ 5000 over 10 years. She must decide betu Bank A and Bank B.
Bank AA is offering an interest rate of 9%9 \% compounded semi-annually. 3ank B is offering an interest rate of 7.5%7.5 \% compounded monthly. etermine which option is better for Caroline. How much more will sh in? (5 marks)

Studdy Solution

STEP 1

What is this asking? We've got three totally different problems here: solving an exponential equation, figuring out radioactive decay time, and comparing investment options. Watch out! Don't mix up bases and exponents, remember that decay means the final amount is *less* than the initial amount, and make sure to use the right compounding periods for the interest calculations!

STEP 2

1. Solve the Exponential Equation
2. Calculate the Decay Time
3. Compare Investment Options

STEP 3

Alright, let's **rewrite** the equation 92x+1=(127)x19^{2x+1} = (\frac{1}{27})^{x-1} so both sides have the same **base**.
Notice that both 99 and 2727 are powers of 33. 99 is 323^2 and 2727 is 333^3.

STEP 4

Let's **substitute** these into our equation: (32)2x+1=(133)x1(3^2)^{2x+1} = (\frac{1}{3^3})^{x-1}.
Remember 133\frac{1}{3^3} is the same as 333^{-3}.
So, our equation becomes 32(2x+1)=(33)x13^{2(2x+1)} = (3^{-3})^{x-1}.

STEP 5

Now, we **multiply the exponents**: 34x+2=33(x1)3^{4x+2} = 3^{-3(x-1)}, which simplifies to 34x+2=33x+33^{4x+2} = 3^{-3x+3}.
Since the **bases are equal**, the **exponents must be equal** too!

STEP 6

So, we have the equation 4x+2=3x+34x+2 = -3x+3.
Let's **add** 3x3x to both sides to get 7x+2=37x+2 = 3.
Then, **subtract** 22 from both sides: 7x=17x = 1.

STEP 7

Finally, **divide** both sides by 77 to find x=17x = \frac{1}{7}.
Boom!

STEP 8

Hold up!
The problem says the element decays *from* 88g *to* 1010g.
That's growth, not decay!
It should be from 1010g to 88g.
Let's **fix** that.
Our given formula is A=A0(12)thA = A_0(\frac{1}{2})^{\frac{t}{h}}.

STEP 9

Here, AA is the **final amount** (88g), A0A_0 is the **initial amount** (1010g), tt is the **time** we want to find, and hh is the **half-life** (4545 years).
Let's **plug** in the values: 8=10(12)t458 = 10(\frac{1}{2})^{\frac{t}{45}}.

STEP 10

Let's **isolate** the exponential term by **dividing** both sides by 1010: 810=(12)t45\frac{8}{10} = (\frac{1}{2})^{\frac{t}{45}}, which simplifies to 45=(12)t45\frac{4}{5} = (\frac{1}{2})^{\frac{t}{45}}.

STEP 11

To solve for tt, we can use **logarithms**.
Taking the logarithm base 12\frac{1}{2} of both sides gives us log12(45)=t45\log_{\frac{1}{2}}(\frac{4}{5}) = \frac{t}{45}.

STEP 12

Now, **multiply** both sides by 4545 to get t=45log12(45)t = 45 \cdot \log_{\frac{1}{2}}(\frac{4}{5}).
Using a calculator, we find t15.47t \approx 15.47 years.

STEP 13

For **Bank A**, the interest rate is 9%9\% compounded semi-annually.
So, the **interest rate per compounding period** is 0.092=0.045\frac{0.09}{2} = 0.045, and the **number of compounding periods** over 1010 years is 102=2010 \cdot 2 = 20.

STEP 14

Using the compound interest formula A=P(1+r)nA = P(1 + r)^n, where PP is the **principal** ($5000\$5000), rr is the **interest rate per period**, and nn is the **number of periods**, we get A=5000(1+0.045)20$12,045.07A = 5000(1 + 0.045)^{20} \approx \$12,045.07.

STEP 15

For **Bank B**, the interest rate is 7.5%7.5\% compounded monthly.
The **interest rate per period** is 0.07512=0.00625\frac{0.075}{12} = 0.00625, and the **number of periods** is 1012=12010 \cdot 12 = 120.

STEP 16

Using the same formula, we get A=5000(1+0.00625)120$10,568.61A = 5000(1 + 0.00625)^{120} \approx \$10,568.61.

STEP 17

Bank A is the better option.
Caroline will earn $12,045.07$10,568.61=$1476.46\$12,045.07 - \$10,568.61 = \$1476.46 more with Bank A.

STEP 18

1. x=17x = \frac{1}{7}
2. It will take approximately 15.4715.47 years for 1010g to decay to 88g.
3. Bank A is better; Caroline will earn $1476.46\$1476.46 more.

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