Respuesta :
V = kl where k is the constant of proportionality
V = 24l = 24*30 = 720 in^3
720 = 30*h*w
hw = 720 / 30 = 24
also 2h + 2w = 20
w + h = 10
24/h + h = 10
24 + h^2 = 10h
h^2 - 10h + 24 = 0
(h - 4)(h - 6)=0
h = 4 inches or 6 inches.
V = 24l = 24*30 = 720 in^3
720 = 30*h*w
hw = 720 / 30 = 24
also 2h + 2w = 20
w + h = 10
24/h + h = 10
24 + h^2 = 10h
h^2 - 10h + 24 = 0
(h - 4)(h - 6)=0
h = 4 inches or 6 inches.
Answer:
Volume of box (V) is given by:
[tex]V = lwh[/tex] .....[1]
where,
l is the length , w is the width and h is the height of the box respectively.
As per the statement:
The volume of a box(V) varies directly with its length(l).
[tex]V \propto l[/tex]
then;
[tex]V = kl[/tex] where, k is the constant of proportionality.
Substitute k = 24 and l = 30 inches we have;
[tex]V = 24 \cdot 30 = 720 in^3[/tex]
Substitute the given values of V and l in [1] we have;
[tex]720 = 30wh[/tex]
Divide both sides by 30h we have;
[tex]\frac{24}{h} =w[/tex] .....[2]
It is also given that the girth of 20 inches (perimeter of the side formed by the width and height)
Perimeter of rectangle formed by width and height is given by:
P = 2(w+h)
then;
[tex]20 = 2(w+h)[/tex]
Divide both sides by 2 we have;
[tex]10 = w+h[/tex]
or
w+h = 10 .....[3]
Substitute equation [2] into [3] we have;
[tex]\frac{24}{h}+h = 10[/tex]
⇒[tex]24 +h^2 = 10h[/tex]
⇒[tex]h^2-10h+24 = 0[/tex]
⇒[tex]h^2-6h-4h+24=0[/tex]
⇒[tex]h(h-6)-4(h-6)=0[/tex]
Take h-6 common we have;
⇒[tex](h-6)(h-4)=0[/tex]
By zero product property we have;
h-6=0 or h-4=0
⇒h = 6 inches or h = 4 inches.
therefore, the height is, 6 or 4 inches
