Question Find dimensions of a rectangle where length is $11 \mathrm{yd}$ less than 3 times the width, and area is $70 \mathrm{yd}^{2}$ .

Studdy Solution

STEP 1

Assumptions
1. Let the width of the rectangle be $w$ yards.
2. The length of the rectangle is $3w - 11$ yards.
3. The area of the rectangle is $70$ square yards.
4. The area of a rectangle is given by the formula $\text{Area} = \text{length} \times \text{width}$ .

STEP 2

Use the formula for the area of a rectangle to set up an equation with the given information.
$\text{Area} = \text{length} \times \text{width}$

STEP 3

Substitute the expressions for the length and the area into the area formula.
$70 = (3w - 11) \times w$

STEP 4

Distribute $w$ to both terms inside the parentheses.
$70 = 3w^2 - 11w$

STEP 5

Rearrange the equation to form a quadratic equation by moving all terms to one side.
$3w^2 - 11w - 70 = 0$

STEP 6

Factor the quadratic equation, if possible, to find the values of $w$ .

STEP 7

To factor the quadratic equation, look for two numbers that multiply to $3 \times (-70) = -210$ and add up to $-11$ .

STEP 8

The numbers that satisfy these conditions are $-21$ and $10$ .

STEP 9

Rewrite the middle term of the quadratic equation using the numbers found in STEP_8.
$3w^2 - 21w + 10w - 70 = 0$

STEP 10

Factor by grouping. Group the first two terms together and the last two terms together.
$(3w^2 - 21w) + (10w - 70) = 0$

STEP 11

Factor out the greatest common factor from each group.
$3w(w - 7) + 10(w - 7) = 0$

STEP 12

Factor out $(w - 7)$ which is common in both terms.
$(w - 7)(3w + 10) = 0$

STEP 13

Set each factor equal to zero to solve for $w$ .
$w - 7 = 0 \quad \text{or} \quad 3w + 10 = 0$

STEP 14

Solve the first equation for $w$ .
$w = 7$

STEP 15

Solve the second equation for $w$ .
$3w = -10$
$w = -\frac{10}{3}$

STEP 16

Since a width cannot be negative, we discard $w = -\frac{10}{3}$ and keep $w = 7$ yards as the width of the rectangle.

STEP 17

Now that we have the width, we can find the length by using the expression $3w - 11$ .

STEP 18

Substitute $w = 7$ into the expression for the length.
$\text{length} = 3(7) - 11$

STEP 19

Calculate the length.
$\text{length} = 21 - 11$

STEP 20

Find the length of the rectangle.
$\text{length} = 10 \text{ yards}$
The dimensions of the rectangle are a width of 7 yards and a length of 10 yards.

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms