Please help me answer this question;
1.A metal gutter open at the tops and ends, is bent up from material 30 cm wide to form a rectangular cross section.
a) If the base of the gutter is x cm wide, find the height of the gutter.
b)Show that the area of the gutter cross section is given by A = 15x - x^2/2.
Thanks.
Make a sketch of the cross-section
I would assume the you are bending the 30 cm wide piece so that the two sides of the gutter are of equal height.
then if the base is x cm, then the height must be (30-x)/2 or 15 - x/2 or 15 - (1/2)x
Area = base x height
= x(15 - x/2) = 15x - x^2/2
thank you very much for your help.
To answer the question, let's break it down step by step.
a) If the base of the gutter is x cm wide, find the height of the gutter.
To find the height of the gutter, we need to use the information given about the gutter being bent up from material that is 30 cm wide.
Let's imagine the flat sheet of metal before it is bent. The width of the sheet is 30 cm, and when it is bent, it forms a rectangular cross section with a base width of x cm.
Since the material is bent up, the width of the base will become the height of the gutter. Therefore, the height of the gutter is x cm.
So, the height of the gutter is x cm.
b) Show that the area of the gutter cross section is given by A = 15x - x^2/2.
To find the area of the gutter cross section, we need to multiply the base width (x cm) by the height (x cm).
Area = base × height = x × x = x^2
However, we need to express the area in terms of x and simplify it further.
Given that the material is 30 cm wide, we know that the total width (including the two sides bent up) is 30 cm. So, we have the equation:
x + 2(x) = 30
x + 2x = 30
3x = 30
x = 10
Now, substitute the value of x into the area equation:
A = x^2 = (10)^2 = 100
Simplifying further, we get:
A = 15x - x^2/2
= 15(10) - (10)^2/2
= 150 - 100/2
= 150 - 50
= 100
So, the area of the gutter cross section is indeed given by A = 15x - x^2/2, which simplifies to 100 when x equals 10.
Therefore, we have shown that the area of the gutter cross section is given by A = 15x - x^2/2.