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The product of two consecutive odd integers is 99. Find the integers.
Two consecutive odd numbers whose product is 99 are 9 and 11, and and .
The product of two consecutive odd integers is 168. Find the integers.
Two consecutive even numbers whose product is 168 are 12 and 14,and and .
We will use the formula for the area of a triangle to solve the next example.
For a triangle with base and height , the area, , is given by the formula .
Recall that, when we solve geometry applications, it is helpful to draw the figure.
An architect is designing the entryway of a restaurant. She wants to put a triangular window above the doorway. Due to energy restrictions, the window can have an area of 120 square feet and the architect wants the width to be 4 feet more than twice the height. Find the height and width of the window.
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Step 1. Read the problem.
Draw a picture. |
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| Step 2. Identify what we are looking for. | We are looking for the height and width. |
| Step 3. Name what we are looking for. | Let
the height of the triangle.
the width of the triangle |
| Step 4. Translate. | We know the area. Write the formula for the area of a triangle.
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| Step 5. Solve the equation. Substitute in the values. |
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| Distribute. |
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| This is a quadratic equation, rewrite it in standard form. |
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| Solve the equation using the Quadratic Formula. Identify the a, b, c values. |
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| Write the quadratic equation. |
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| Then substitute in the values of a, b, c. . |
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| Simplify. |
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| Simplify the radical. |
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| Rewrite to show two solutions. |
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| Simplify. |
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| Since h is the height of a window, a value of does not make sense. |
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| The height of the triangle:
The width of the triangle: | |
| Step 6. Check the answer. Does a triangle with a height 10 and width 24 have area 120? Yes. | |
| Step 7. Answer the question. | The height of the triangular window is 10 feet and the width is 24 feet. |
Notice that the solutions were integers. That tells us that we could have solved the equation by factoring.
When we wrote the equation in standard form, , we could have factored it. If we did, we would have solved the equation .
Find the dimensions of a triangle whose width is four more than six times its height and has an area of 208 square inches.
The height of the triangle is 8 inches and the width is 52 inches.
If a triangle that has an area of 110 square feet has a height that is two feet less than twice the width, what are its dimensions?
The height of the triangle is 20 feet and the width is 11 feet.
In the two preceding examples, the number in the radical in the Quadratic Formula was a perfect square and so the solutions were rational numbers. If we get an irrational number as a solution to an application problem, we will use a calculator to get an approximate value.
The Pythagorean Theorem gives the relation between the legs and hypotenuse of a right triangle. We will use the Pythagorean Theorem to solve the next example.
In any right triangle, where and are the lengths of the legs and is the length of the hypotenuse,
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