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The width of a rectangle is 7in less than the length. The area of the painting is 98in squared. Find the length and width

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Answer:

Length = 14 in.

Width = 7 in.

Explanation:

The formula for the area of a rectangle is given by:

A = lw, where

  • A is the area in units squared,
  • l is the length,
  • and w is the width.

Since the width is 7 in. less than the length, we can show this with the following equation:

w = l - 7

Now we can substitute l - 7 for w in the area formula and 98 for A to find l, the length of the rectangle:

Step 1: Distribute the l:

98 = l(l - 7)

98 = l^2 - 7l

Step 2: Subtract l^2 from both sides:

We have a quadratic and in order to solve, we'll need to put it in standard form, whose general equation is given by:

ax^2 + bx + c = 0

Thus, we'll need to move l^2 and -7l to the left-hand side of the equation by first subtracting l^2 from both sides:

(98 = l^2 - 7l) - l^2

-l^2 + 98 = -7l

Step 3: Add 7l to both sides to put the quadratic in standard form:

Now we can add 7l to both sides to put the quadratic in standard form:

(-l^2 + 98 = -7l) + 7l

-l^2 + 7l + 98 = 0

Step 4: Factor the quadratic:

  • Since the quadratic is in standard form, -1 is our a value, 7 is our b value, and 98 is our c value.
  • We can factor the quadratic by finding two numbers whose product equals a * c (-1 * 98) and whose sum is b (7).
  • -7 and 14 meets these two requirements as -7 * 14 = 98 and -7 + 14 = 7.
  • When putting the quadratic in factored form, we use the opposite sign of -7 and 14.

Thus, the quadratic in factored form is (l + 7)(l - 14) = 0

Step 5: Solve the quadratic by setting each term equal to 0:

  • Since our variable is l, solving the quadratic will give us the length of the rectangle.

We can solve the quadratic by setting (l + 7) and (l - 14) equal to 0:

Setting (l + 7) equal to 0:

(l + 7 = 0) - 7

l = -7

Setting (l - 14) equal to 0:

(l - 14 = 0) + 14

l = 14

Because we can't have a negative side length, the length is 14 in.

Step 6: Identify the width:

  • The only number which you can multiply with 14 to get 98 is 7.

Thus, the width is 7 in.

User Santrack
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