Question

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1. determine the total area of the material needed to create the wedge tent shown. include the material needed for the floor. express your answer to the nearest square foot.

Explanation:

Step1: Identify the tent's components

The wedge tent (a triangular prism - like structure) has: two triangular ends, two rectangular side panels, and a rectangular floor. Wait, actually, looking at the diagram (assuming the tent has a triangular front/back, two slanted sides, and a floor). Let's parse the dimensions: from the diagram, the base of the triangle (front) is 6 ft (I think, maybe the floor length is 6 ft, the triangle's base is 6 ft, height of triangle is 7 ft? Wait, the slant side is 7.8 ft? Wait, maybe the tent is a wedge with a triangular face (base 6 ft, height 7 ft), two rectangular sides (one with dimensions 7 ft by 7.8 ft, another? Wait, maybe the tent has:

- Floor: rectangle with length 6 ft and width 7 ft? Wait, no, let's re - examine. The diagram shows: the front triangle has base 6 ft, height 7 ft? Wait, the yellow part is a rectangle? Wait, maybe the tent is composed of:

1. Two triangular faces (front and back): each with base \( b = 6\) ft and height \( h=7\) ft. Area of one triangle: \( A_{triangle}=\frac{1}{2}\times b\times h\)
2. Two rectangular side panels: one with dimensions \( 7\) ft (height of triangle? No, wait, the slant edge is 7.8 ft. Wait, maybe the tent has a floor (rectangle: length 6 ft, width 7 ft), two triangular sides? No, maybe the correct breakdown is:

Wait, the wedge tent (also called a triangular prism tent) has:

- Two congruent triangular faces (let's say base \( b = 6\) ft, height \( h = 7\) ft)
- Two congruent rectangular faces (with length equal to the length of the tent, say \( l=7\) ft, and width equal to the slant height of the triangle? Wait, no, the slant side of the triangle: using Pythagoras, if the base of the triangle is 6 ft (so half - base is 3 ft), height is 7 ft, then the slant side (hypotenuse) \( s=\sqrt{3^{2}+7^{2}}=\sqrt{9 + 49}=\sqrt{58}\approx7.62\) ft, but the diagram says 7.8 ft. Maybe the base is 6 ft, the other side of the triangle is 7 ft, and the slant side is 7.8 ft. Wait, maybe the tent is made of:

- Floor: rectangle with length 6 ft and width 7 ft. Area \( A_{floor}=6\times7 = 42\) sq ft.
- Two triangular sides: each with base 7 ft and height? Wait, no, let's look at the given dimensions: the diagram has 7 ft (a side), 7.8 ft (slant side), 6 ft (base of the triangle).

Wait, maybe the correct components are:

1. Two triangular faces: each with base \( b = 6\) ft and height \( h = 7\) ft. Area of one triangle: \( A_{triangle}=\frac{1}{2}\times6\times7=21\) sq ft. Two triangles: \( 2\times21 = 42\) sq ft.
2. Two rectangular side panels: one with dimensions \( 7\) ft (length) and \( 7.8\) ft (width), and another with dimensions \( 7\) ft (length) and \( 7\) ft (width)? No, that doesn't make sense. Wait, maybe the tent has a floor (6 ft by 7 ft), two slanted sides (each is a rectangle: 7 ft by 7.8 ft), and a front/back triangle? No, I think I made a mistake. Let's re - approach.

Wait, the wedge tent (a type of triangular prism) has:

- Lateral faces: two rectangles (the slanted sides) and one rectangle (the floor). Wait, no, a triangular prism has two triangular bases and three rectangular lateral faces. But in a tent, maybe the floor is one of the rectangular faces, and the other two are the slanted sides.

Wait, let's assume the triangular base has sides: base \( b = 6\) ft, height \( h = 7\) ft, and the length of the tent (the distance between the two triangular bases) is \( l = 7\) ft. The slant side of the triangle (the hypotenuse) can be calculated as \( s=\sqrt{(\frac{6}{2})^{2}+7^{2}}=\sqrt{9 + 49}=\sqrt{58}\approx7.62\) ft, but the diagram shows 7.8 ft. Maybe the triangular base has sides 6 ft, 7 ft, and 7.8 ft (since \( 6^{2}+7^{2}=36 + 49 = 85\), \( 7.8^{2}=60.84\), no, that's not a right triangle. Wait, maybe the triangle is isoceles with base 6 ft and equal sides 7.8 ft? Then the height of the triangle \( h=\sqrt{7.8^{2}-3^{2}}=\sqrt{60.84 - 9}=\sqrt{51.84}=7.2\) ft. But the diagram has 7 ft. Maybe the given dimensions are:

- Floor: length 6 ft, width 7 ft. Area \( A_{floor}=6\times7 = 42\)
- Two triangular sides: each with base 7 ft and height 6 ft? No. Wait, the problem says "wedge tent", which is a triangular prism. The surface area of a triangular prism is \( 2\times(\frac{1}{2}\times b\times h)+(a + b + c)\times l\), where \( a,b,c\) are the sides of the triangle and \( l\) is the length of the prism.

Assuming the triangular base has sides: \( a = 7\) ft, \( b = 6\) ft, \( c = 7.8\) ft, and the length of the prism (the distance along the tent) is \( l = 7\) ft.

First, calculate the area of the two triangular bases: \( 2\times(\frac{1}{2}\times b\times h)\). Wait, if we take the base of the triangle as 6 ft and the height as 7 ft (from the diagram's 7 ft label), then area of one triangle is \( \frac{1}{2}\times6\times7 = 21\), two triangles: \( 2\times21=42\).

Then, the lateral surface area: the perimeter of the triangle times the length of the prism. The perimeter of the triangle is \( 7 + 6+7.8=20.8\) ft. Lateral surface area: \( 20.8\times7 = 145.6\) sq ft.

But wait, the floor is part of the lateral surface? No, in a tent, the floor is one of the rectangular faces. Wait, maybe the tent's surface area (including the floor) is the sum of the two triangular faces, two slanted rectangular faces, and the floor (rectangular face).

Wait, let's re - define:

- Triangular faces (front and back): each with base 6 ft and height 7 ft. Area of one: \( \frac{1}{2}\times6\times7 = 21\), two: \( 42\)
- Slanted side 1: rectangle with dimensions 7 ft (length) and 7.8 ft (width). Area: \( 7\times7.8 = 54.6\)
- Slanted side 2: rectangle with dimensions 7 ft (length) and 7 ft (width). Area: \( 7\times7 = 49\)
- Floor: rectangle with dimensions 6 ft (length) and 7 ft (width). Area: \( 6\times7 = 42\)

Wait, this is confusing. Maybe the correct way is: the tent is a triangular prism with triangular base (base = 6 ft, height = 7 ft) and length = 7 ft, and the other two sides of the triangle are 7 ft and 7.8 ft.

Wait, let's check the surface area formula for a triangular prism: \( SA=2\times(\frac{1}{2}\times b\times h)+(a + b + c)\times l\), where \( b\) is the base of the triangle, \( h\) is the height of the triangle, \( a\) and \( c\) are the other two sides, and \( l\) is the length of the prism.

So, \( b = 6\), \( h = 7\), \( a = 7\), \( c = 7.8\), \( l = 7\)

First, area of the two triangular bases: \( 2\times(\frac{1}{2}\times6\times7)=42\)

Perimeter of the triangle: \( 7 + 6+7.8 = 20.8\)

Lateral surface area: \( 20.8\times7=145.6\)

Total surface area: \( 42 + 145.6=187.6\approx188\) sq ft. Wait, but maybe the floor is the base \( b\times l=6\times7 = 42\), and the two slanted sides are \( a\times l\) and \( c\times l\), and the two triangular faces. Wait, no, the formula \( 2\times(\frac{1}{2}\times b\times h)+(a + b + c)\times l\) includes all faces: two triangles and three rectangles (the three sides of the triangle times the length). So that should be the total surface area including the floor (since the floor is the rectangle with side \( b\) and length \( l\)).

So let's recalculate:

- Area of two triangles: \( 2\times\frac{1}{2}\times6\times7 = 42\)
- Area of rectangle with side \( a = 7\) and length \( l = 7\): \( 7\times7 = 49\)
- Area of rectangle with side \( c = 7.8\) and length \( l = 7\): \( 7.8\times7 = 54.6\)
- Area of rectangle with side \( b = 6\) and length \( l = 7\): \( 6\times7 = 42\)

Now sum them up: \( 42+49 + 54.6+42=187.6\approx188\) square feet.

Wait, maybe I misread the dimensions. Let's assume the triangular base has base 6 ft, height 7 ft, and the length of the tent (the distance along the tent) is 7 ft, and the other two sides of the triangle are 7 ft and 7.8 ft. Then the total surface area is the sum of the two triangles, and the three rectangles (the three sides of the triangle times the length of the tent).

So:

Step1: Calculate area of two triangular bases

The formula for the area of a triangle is \( A=\frac{1}{2}\times base\times height\). For one triangle, \( base = 6\) ft, \( height = 7\) ft. So area of one triangle is \( \frac{1}{2}\times6\times7 = 21\) sq ft. For two triangles, \( 2\times21 = 42\) sq ft.

Step2: Calculate area of three rectangular faces

- Rectangle 1: side \( a = 7\) ft, length \( l = 7\) ft. Area \( A_1=7\times7 = 49\) sq ft.
- Rectangle 2: side \( c = 7.8\) ft, length \( l = 7\) ft. Area \( A_2=7.8\times7 = 54.6\) sq ft.
- Rectangle 3: side \( b = 6\) ft, length \( l = 7\) ft. Area \( A_3=6\times7 = 42\) sq ft.

Step3: Sum all areas

Total area \(=42 + 49+54.6 + 42=187.6\approx188\) sq ft.

Answer:

\(188\) square feet (rounded to the nearest square foot)