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A direct variation includes the points $(4,16)$ and $(1,n)$ . Find $n$ . Write and solve a direct variation equation to find the answer. $\newline$ $n = \_\_\_$

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Write and solve direct variation equations

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Q. A direct variation includes the points

(4,16)

and

(1,n)

. Find

n

. Write and solve a direct variation equation to find the answer.

\newline

n = \_\_\_

Select Equation: Select the equation that represents direct variation. $\newline$ y is directly proportional to $x$ . $\newline$ Direct variation: $y = kx$
Substitute Values: We know that $y = 16$ when $x = 4$ . Substitute the values in $y = kx$ . Plug in $4$ for $x$ and $16$ for $y$ in $y = kx$ . $16 = k \times 4$
Find Constant: Now let's find the constant of proportionality $k$ . $\newline$ Divide both sides by $4$ . $\newline$ $\frac{16}{4} = \frac{(k \times 4)}{4}$ $\newline$ $4 = k$ $\newline$ $k = 4$
Write Equation: We have: $\newline$ $k = 4$ $\newline$ Write the direct variation equation $y = kx$ using the value of $k$ . $\newline$ Substitute $4$ for $k$ in $y = kx$ . $\newline$ $y = 4x$
Find $y$ : Direct variation equation: $y = 4x$ $\newline$ Find $y$ when $x = 1$ . $\newline$ Substitute $1$ for $x$ in $y = 4x$ . $\newline$ $y = 4(1)$ $\newline$ $y = 4$

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10-\mathrm{kg}

bucket is lifted from the ground to a height of

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. Initially the bucket contains

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10\text{-kg}

bucket is lifted from the ground to a height of

14\text{ m}

at a constant speed with a rope that weighs

0.6\text{ kg/m}

. Initially the bucket contains

42\text{ kg}

of water, but the water leaks at a constant rate and finishes draining just as the bucket reaches the

14\text{-m}

level. How much work is done? (Use

9.8\text{ m/s}^2

g

.) Show how to approximate the required work (in

\text{J}

) by a Riemann sum. (Let

x

be the height in meters above the ground. Enter

x_i^*

x_i

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