Chapter 7.1, Problem 24IP

Solution

(a)

Use the Distributive property to write $12\left(24-h\right)$ as an equivalent algebraic expression.

  $288-12h$

Given information:

The expression, $12\left(24-h\right)$

Concept used:

• The distributive property of binomials (FOIL): $\left(a+b\right)\left(c+d\right)=a\cdot c+a\cdot d+b\cdot c+b\cdot d$
• Left Distributive property: $a\left(b+c\right)=a\cdot b+a\cdot c$
• Right Distributive property: $\left(a\pm b\right)c=a\cdot c\pm b\cdot c$
• Exponent property: $x^n\cdot x^m=x^{n+m}$

Calculation:

In order to simplify the given expression $12\left(24-h\right)$ , first use the distributive property to expand the parenthesis and then simplify further by using the exponent property as shown below

  $\begin{array}{c}12\left(24-h\right)=12\left(24-h\right)\\ =12\left(24\right)+12\left(-h\right)\\ =288-12h\end{array}$

Thus, the given expression simplifies to $288-12h$ .

(b)

Make a table of ordered pairs of (h,v) of the equation $288-12h$ .

h	v	(v,h)
0	288	(0,288)
1	276	(1,276)
2	264	(2,264)
3	252	(3,252)
4	240	(4,240)

Given information:

The expression, $288-12h$

Explanation:

In order to make a table for v,h by using the expression $v=288-12h$ ,putting any value of h to get the value of v. So, the table is :

h	v	(v,h)
0	288	(0,288)
1	276	(1,276)
2	264	(2,264)
3	252	(3,252)
4	240	(4,240)

(c)

Graph the ordered pairs on the coordinate plane.

GIVEN:

The expression, $12\left(24-h\right)$

Concept Used:

In rectangular coordinate system any point be in the form of $\left(x,y\right)$ .

In Quadrant I, both the coordinates of ordered pair are positive, that is, $\left(+,+\right)$ .

In Quadrant II, first number is negative and second is positive, that is, $\left(-,+\right)$ .

In Quadrant III, both the coordinates of ordered pair are negative, that is, $\left(-,-\right)$ .

In Quadrant IV, first point is positive and second is negative, that is, $\left(+,-\right)$ .

Explanation: In order to plot the point $\left(0,288\right)$ , observe that both numbers of the ordered pair are positive, so the point will be in first quadrant.To plot the graph of the point, first move 0 units to the right starting from origin and then move 288 units up.

In order to plot the point $\left(1,276\right)$ , observe that both numbers of the ordered pair are positive, so the point will be in first quadrant.To plot the graph of the point, first move 1 units to the right starting from origin and then move 276 units up.

In order to plot the point $\left(2,264\right)$ , observe that both numbers of the ordered pair are positive, so the point will be in first quadrant.To plot the graph of the point, first move 2 units to the right starting from origin and then move 264 units up.

In order to plot the point $\left(3,252\right)$ , observe that both numbers of the ordered pair are positive, so the point will be in first quadrant.To plot the graph of the point, first move 3 units to the right starting from origin and then move 252 units up.

In order to plot the point $\left(4,240\right)$ , observe that both numbers of the ordered pair are positive, so the point will be in first quadrant.To plot the graph of the point, first move 4 units to the right starting from origin and then move 240 units up.

The graph of the point is shown below:

(d)

Explain what happen to the volume as the height increases.

Given information:

The expression, $v=12\left(24-h\right)$

Explanation:

If the height of the volume increases then there volume will be decreases because of the equation given $v=12\left(24-h\right)$ .