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3 years ago

Write an equation for the exponential function represented in the table and graph below.

Mathematics

1 answer:

k0ka [10]3 years ago

8 0

<u>Given</u>:

Given that the values of x and y in the table.

We need to determine the equation for the exponential function represented in the table and graph the equation.

<u>Equation for the exponential function:</u>

The general equation for the exponential function is given by

$y=a(b)^x$ ------- (1)

Substituting the coordinate (0,6) in equation (1), we get;

$6=a(b)^0$

$6=a$

Thus, the value of a is 6.

Substituting a = 6 in equation (1), we have;

$y=6(b)^x$ -------- (2)

Substituting the coordinate (1,12) in equation (2), we get;

$12=6(b)^1$

$12=6b$

$2=b$

Thus, the value of b is 2.

Substituting b = 2 in equation (2), we get;

$y=6(2)^x$

Thus, the equation for the exponential function is $y=6(2)^x$

The image of the graph of the exponential function is attached below:

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The two shorter sides of a triangle are the same length. The length of the longer side is 5 m longer than each of the shorter si

Novosadov [1.4K]

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3 years ago

Use mathematical induction to prove the statement is true for all positive integers n, or show why it is false:

kondaur [170]

$\text{Proof by induction:}$
$\text{Test that the statement holds or n = 1}$

$LHS = (3 - 2)^{2} = 1$
$RHS = \frac{6 - 4}{2} = \frac{2}{2} = 1 = LHS$
$\text{Thus, the statement holds for the base case.}$

$\text{Assume the statement holds for some arbitrary term, n= k}$
$1^{2} + 4^{2} + 7^{2} + ... + (3k - 2)^{2} = \frac{k(6k^{2} - 3k - 1)}{2}$

$\text{Prove it is true for n = k + 1}$
$RTP: 1^{2} + 4^{2} + 7^{2} + ... + [3(k + 1) - 2]^{2} = \frac{(k + 1)[6(k + 1)^{2} - 3(k + 1) - 1]}{2} = \frac{(k + 1)[6k^{2} + 9k + 2]}{2}$

$LHS = \underbrace{1^{2} + 4^{2} + 7^{2} + ... + (3k - 2)^{2}}_{\frac{k(6k^{2} - 3k - 1)}{2}} + [3(k + 1) - 2]^{2}$
$= \frac{k(6k^{2} - 3k - 1)}{2} + [3(k + 1) - 2]^{2}$
$= \frac{k(6k^{2} - 3k - 1) + 2[3(k + 1) - 2]^{2}}{2}$
$= \frac{k(6k^{2} - 3k - 1) + 2(3k + 1)^{2}}{2}$
$= \frac{k(6k^{2} - 3k - 1) + 18k^{2} + 12k + 2}{2}$
$= \frac{k(6k^{2} - 3k - 1 + 18k + 12) + 2}{2}$
$= \frac{k(6k^{2} + 15k + 11) + 2}{}$
$= \frac{(k + 1)[6k^{2} + 9k + 2]}{2}$
$= \frac{(k + 1)[6(k + 1)^{2} - 3(k + 1) - 1]}{2}$
$= RHS$

Since it is true for n = 1, n = k, and n = k + 1, by the principles of mathematical induction, it is true for all positive values of n.

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3 years ago

If you can walk 4.9 miles in 2.9 hours, how many minutes does it take to walk 8.73 miles? Round to 1 decimal.

gavmur [86]

Step-by-step explanation:

2.9 hours = 174mins

174 mins = 4.9miles

1 miles = 1740/49 mins

8.73 miles = 310.0mins (1dp)

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