Please explain how to set this up and how to solve:
A school's rectangular athletic fields currently have a length of 125 yd and a width of 75 yd. the school plans to expand both the length and the width of the fields by x yards. What polynomial is standard form represents the area of the expanded athletic field?
A = Area
L = Lenght
W = Width
New lenght L = 125 + x
New width W = 75 + x
A = W * L
A = ( 125 + x ) * ( 75 + x )
A = 75 * 125 + 75 * x + x * 125 + x * x
A = 9375 + 75 x + 125 x + x ^ 2
A = x ^ 2 + 200 x + 9375
I loved the way you broke it down and explained it to me in detail. What a great help!
(x+125)(x+75) = Area
Well, if I were the school, I'd probably hire a team of elephants to expand the fields by x yards. They have those big trunks, you know? But I digress.
To find the polynomial in standard form that represents the area of the expanded athletic field, we need to multiply the new length and width. So, the new length would be (125 + x) yards, and the new width would be (75 + x) yards.
Now, to find the area, we simply multiply the two lengths: (125 + x) * (75 + x).
To put it in standard form, you'll need to multiply everything out and combine similar terms. It might look a bit messy at first, but with a little bit of patience and some math skills, you'll get there. Good luck!
To determine the polynomial in standard form that represents the area of the expanded athletic field, we need to follow these steps:
Step 1: Find the area of the current field
The area of the rectangular field is given by multiplying its length by its width.
Area = Length × Width = 125 yd × 75 yd
Step 2: Expand the dimensions
To find the dimensions of the expanded field, we add x yards to both the length and the width.
Expanded Length = Length + x = 125 yd + x
Expanded Width = Width + x = 75 yd + x
Step 3: Calculate the area of the expanded field
The area of the expanded field will be the product of the expanded length and expanded width.
Expanded Area = Expanded Length × Expanded Width = (125 yd + x) × (75 yd + x)
Step 4: Multiply binomials
To simplify the expanded area, we need to multiply the binomials using the distributive property.
Expanded Area = 125 yd × 75 yd + 125 yd × x + 75 yd × x + x × x
Step 5: Simplify the polynomial
Combine like terms to simplify the polynomial expression:
Expanded Area = 9375 yd² + 125 yd x + 75 yd x + x²
Step 6: Arrange in standard form
To write the polynomial in standard form, arrange the terms in decreasing order of the exponent.
Expanded Area = x² + 125 yd x + 75 yd x + 9375 yd²
Thus, the polynomial in standard form that represents the area of the expanded athletic field is:
f(x) = x² + 200 yd x + 9375 yd²