Question

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The curve y = ax^n, where a and n are constants, passes through the points (2.25, 27), (4, 64) and (6.25, p). Without using logarithms, calculate the value of a, of n and of p.​

Answer 1

a. The value of n is 3/2

b. The value of a is 8.

c. The value of p is 125

The curve is an exponential function

What is an exponential function?

An exponential function is a function of the form y = axⁿ where a and n are constants

a. How to find the value of n?

Since y = axⁿ where a and n are constants, passes through the points (2.25, 27), (4, 64) and (6.25, p), substituting these points into the equation, we have

y = axⁿ

27 = a(2.25)ⁿ  (1)

64 = a(4)ⁿ        (2)

P = a(6.25)ⁿ     (3)

Dividing (2) by (1), we have

64/27 = a(4)ⁿ/a(2.25)ⁿ

4³/3³ = (4 ÷ 2¹/₄)ⁿ

4³/3³ = (4 ÷ ⁹/₄)ⁿ

4³/3³ = (4 ×  4/9)ⁿ

4³/3³ = (16/9)ⁿ

4³/3³ = (4²/3²)ⁿ

(4/3)³ = (4/3)²ⁿ

Equating exponents, we have

2n = 3

n = 3/2

The value of n is 3/2

b. What is the value of a?

Using equation (2)

64 = a(4)ⁿ        

a = 64/(4)ⁿ        

substituting n = 3/2 into the equation, we have

$a = \frac{64}{4^{\frac{3}{2} } } \\= \frac{64}{(\sqrt{4 } )^{3} } \\= \frac{64}{2^{3} } \\= \frac{64}{8} \\= 8$

So, the value of a is 8.

c. What is the value of p.

Using equation (3), we have

P = a(6.25)ⁿ     (3)

substituting the values of a and n into the equation,we have

$P = 8(6.25)^{\frac{3}{2} } \\= 8(\sqrt{6.25}) ^{3} \\= 8(2.5}) ^{3} \\= 8(15.625})\\= 125$

The value of p is 125

Learn more about exponential function here:

https://brainly.com/question/12940982

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