Answer to Question #260964 in Algebra for bri

Question #260964

(2) Determine k such that kx^2 - 4x + 1/2=0 has real and different solutions. (3) (i) What rate of interest compounded annually is needed to make an investment of $6,500 accumulate to an amount of $7166.25 at the end of two years? (ii) How long will it take for the initial amount to double, compounded annually at the rate in (i) above?


1
Expert's answer
2021-11-11T14:38:55-0500

2)

The polynomial equation is,ย "kx\\\\^2-4x+\\frac{1}{2}=0"

Where "a = k, b= -4, c=\\frac{1}{2}"

Since the equation has distinct root;

therefore "b\\\\^2-4ac > 0"

"(-4)\\\\^2-4\\times k \\times\\frac{1}{2} > 0\\\\\n16 - 2k>0\\\\\n16>2k\\\\\n8>k" , having two real roots,.


3)

i)

Compounded annually formula is "A = P(1 + r)\\\\^t"

A = $7166.25

t = 2

P $6500

Therefore,

"7166.25 = 6500(1 + r)\\\\^2\\\\\n\\frac{7166.25}{6500} = (1 + r)\\\\^2\\\\\n1.1025 = (1 + r)\\\\^2\\\\\n\n\\\\\\sqrt{1.2025} - 1 = r\\\\\nr = 0.050013"

Convert rate of interest as percentage: "R = r \\times 100\\\\"

Rates (R) = 5.002 per year.

ii) The amount double = "2 \\times 6500 = \\$13000\\\\"

A = $13000

Principal = $6,500

Rate of interest = 5.002%

t= t

"13000 = 6500(1 + 0.05002)\\\\ ^ t\\\\\n2 = (1.05002)\\\\ ^ t"

Taking log on both sides:

"\\log2 = t\\log(1.05002)\\\\\nt = \\frac{\\log 2}{\\log 1.05002}\\\\\n14.201 = t"

Time for amount to be doubled is 14 years and 2 months




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