Mathematics, 23.09.2019 02:30, fernandaElizondo

An open top box with a square base has a surface area of 100 square inches. express the volume of the box as a function of the length of the edge of the base. what is its domain?

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Let the base be a square of side length = a ft, and height h ft.

The box has a base of area: $a \cdot a = a^2 (square ft)$ ,

and 4 lateral faces of area ah (square ft).

thus the total area of the box is a^2+4ah

.

we can reduce the number of variables as follows:

$a^2+4ah=100\\\\4ah=100-a^2\\\\h= \frac{100-a^2}{4a}= \frac{25}{a}- \frac{a}{4}$

The volume of the box is given by the function:

$V(a)=a \cdot a \cdot (\frac{25}{a}- \frac{a}{4} )=25a- \frac{a^3}{4}$ .

To find the domain we solve

$25a- \frac{a^3}{4} \ \textgreater \ 0\\\\a(25- ( \frac{a}{2} )^{2} )\ \textgreater \ 0\\\\a(5- \frac{a}{2})(5+ \frac{a}{2} )\ \textgreater \ 0$

the roots of the expression are 0, 10 and -10. Since the overall sign of the expression is negative, we have the following sign table:

++++++++++++(-10)----------(0)++++++++++(10)--------------------

thus the solution of the inequation is (-infinity, -10) union (0, 10)

but we have another condition: a>0, since each side must be positive.

combining these 2 conditions, we have the domain: (0, 10)

$V(a)=25a- \frac{a^3}{4}$ , Domain (0, 10)

answered: brittanydeanlen

The volume = 25x - ¹/₄x³ (in cubic inches)Its domain (0, 10)Further explanation

Given:

An open-top box with a square base has a surface area of 100 square inches.

Question:

Express the volume of the box as a function of the length of the edge of the base. What is its domain?

The Process:

Let the length of the edge of the base = x Let height = h

Part-1: The surface area

Let us arrange the equation to get the surface area of the box with a square base. Recall that the box is without a lid and its surface area is 100 square inches.

$\boxed{ \ Surface \ area = area \ of \ base + (4 \times area \ of \ rectangle) \ }$

$\boxed{ \ x^2 + (4 \cdot x \cdot h) = 100 \ }$

$\boxed{ \ x^2 + 4xh = 100 \ }$

From the above equation, we set it again so that "xh" is the subject on the left.

Both sides are subtracted by x².

$\boxed{ \ 4xh = 100 - x^2 \ }$

Both sides are divided by 4.

$\boxed{ \ xh = \frac{100 - x^2}{4} \ }$ ... (Equation-1)

That is the strategy we have prepared.

- - - - - - - - - -

Part-2: The volume

$\boxed{ \ Volume \ of \ the \ box = length \times width \times height \ }$

$\boxed{ \ Volume = x \cdot x \cdot h \ }$

$\boxed{ \ Volume = x^2h \ }$ ...(Equation-2)

Substitution Equation-1 into Equation-2.

$\boxed{ \ Volume = x(xh) \ }$

$\boxed{ \ Volume = x \bigg( \frac{100 - x^2}{4} \bigg) \ }$

$\boxed{ \ Volume = \frac{100x - x^3}{4} \ }$

$\boxed{ \ Volume = 25x - \frac{1}{4}x^3 \ }$

Thus, an expression of the volume of the box as a function of the length of the edge of the base is $\boxed{\boxed{ \ Volume = 25x - \frac{1}{4}x^3 \ }}$

- - - - - - - - - -

Part-3: The domain of volume

The value of volume must always be positive, i.e., V > 0.

$\boxed{ \ 25x - \frac{1}{4}x^3 0 \ }$

Both sides are multiplied by 4.

$\boxed{ \ 100x - x^3 0 \ }$

Both sides are multiplied by -1, notice the change in the sign of the inequality.

$\boxed{ \ x^3 - 100x < 0 \ }$

$\boxed{ \ x(x^2 - 100) < 0 \ }$

$\boxed{ \ x(x - 10)(x + 10) < 0 \ }$

We get $\boxed{ \ x = 0, \ x = 10, \ and \ x = - 10 \ }$ .

Since the values of x cannot be negative, x = -10 are promptly rejected. For x = 0 can be used as one of the domain limits.

Consider the test of signs:

x(x - 10) (x + 10) is negative to the left of x = 10, and positive to the right of x = 10 on the number line.

Examples of tests:

$\boxed{ \ for \ x = 2 \rightarrow 2(2 - 10)(2 + 10) < 0 \ }$ $\boxed{ \ for \ x = 11 \rightarrow 11(11 - 10)(11 + 10) 0 \ }$

Remember this form above, $\boxed{\ x(x - 10)(x + 10) < 0 \ }$ , the value of the test result must be negative (because < 0).

Thus, the domain of the volume is $\boxed{ \ 0 < x < 10 \ }$ or $\boxed{ \ (0, 10) \ }$

Learn moreFind out the area of a cube linkWhat is the volume of each prism? linkThe volume of a rectangular prism link

Keywords: an open-top box, with, a square base, has a surface area, 100 square inches, express, the volume, as a function, the length, edge, base, what, its domain, test of signs, substitution

An open top box with a square base has a surface area of 100 square inches. express the volume of th

answered: Guest

the system of inequity is

$y\leq -x+4$

$3y> x$

step-by-step explanation:

from the given graph it is noticed that we have two related lines. one is solid and another is dashed.

if a line passing through two points, then the equation of line is

$y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)$

the solid line passing through the points (0,4) and (4,0).

$y-4=\frac{0-4}{4-0}(x-0)$

$y=-x+4$

the related equation of solid line is $y=-x+4$ .

similarly, find the related equation of dashed line. the dashed liune passing through the points (0,0) and (3,1).

$y-0=\frac{0-1}{3-0}(x-0)$

$3y=x$

the related equation of dashed line is $3y=x$

from the given graph it is noticed that the point (0,1) lies on the shaded region, therefore the point (0,1) must be satisfied by the related equations.

check the related equations by the point (0,1).

$1=-0+4$

$1=4$

this statement is true if the sign of inequality is < instead of equality. since it is an solid line therefore the sign of inequality will be $\leq$ . the required inequality is