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Alice is constructing rectangular prisms to represent buildings in a diorama for her social
studies class.
• The length of each rectangular prism exceeds the width by 6 cm.
• The height of each rectangular prism is 5 cm less than the width.
• The volume of each rectangular prism must be less than or equal to 50 cm3.
What are possible width values in centimeters for each rectangular prism?

Respuesta :

tehorsesnameisfriday
l = w + 6

w = l - 6

h = w - 5

h = l - 11

V = lwh

V = l(l - 6)(l - 11)

But V <= 50

therefore,

l^3 - 17l^2 + 66l - 50 <= 0

since all the coefficients add to 0, 1 is a root. Therefore, it can be factored into:

(l-1)(l-8+sqrt(14))(l-8-sqrt(14))<= 0

but at the extremes,

h >= 0

l -11 >= 0

l >= 11

therefore, considering this,

we can see that

11 <= l <= 8 + sqrt(14)

11 - 6 <= l - 6 <= 8 + sqrt(14) - 6

5 <= w <= 2 + sqrt(14)

(In cm)

The possible width values in centimeters for each rectangular prism is;

5 cm and 11 cm

Let the length of prism be l

Let the width of the prism be w

Let the height of the prism be h

We are told that length of each rectangular prism exceeds the width by 6 cm.

Thus;

l = w + 6    ----(eq 1)

We are told that height of each rectangular prism is 5 cm less than the width. Thus;

h = w - 5   ----(eq 2)

We are told that the volume of each rectangular prism must be less than or equal to 50 cm³.

Thus; V ≤ 50

Formuls for volume here is;

V = lwh

plugging in relevant expressions gives;

V = (w + 6) × w × (w - 5)

Expanding this bracket gives;

V = w(w² + w - 30)

V = w³ + w² - 30w

At V = 0, w = 0, 5 or -6

w cannot be zero and so we adopt w= 5

Thus;

l = 5 + 6

l = 11

Now, it is possible that instead of using w = 5 and l = 11, they can switch it to have w = 11 cm and l = 5 cm

In conclusion the possible width values are 5 cm and 11 cm

Read more at; https://brainly.com/question/386916

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