A large painting in the style of Rubens is 3 feet longer than it is wide. If the wooden frame is 18 in wide and the area of the picture frame is 286 ft^2, find the dimensions of the painting.
I have the basic idea down of what I'm supposed to do to solve it.. But I just can't put it all together in my head. Helpppp
Always draw a diagram, then use what info you have been given.
width = w
length = w+3
Since the frame is 1.5' wide, the total width and length get increased by 3 ft for the entire framed picture.
picture area: width*length = w(w+3)
total framed area: (width+3)(length+3) = (w+3)(w+6)
The total area less the area of the picture is the area of the frame:
(w+3)(w+6) - w(w+3) = 286
w^2 + 9w + 18 - w^2 - 3w = 286
6w + 18 = 286
6w = 268
w = 44' 8"
wow! Big painting!
Ah, the world of art and math colliding! Let's tackle this puzzle together, shall we?
First, let's break down the information we have:
1. The wooden frame is 18 inches wide.
2. The area of the picture frame is 286 square feet.
3. The painting is in the style of Rubens and is 3 feet longer than it is wide.
Now, let's find a way to piece it all together:
Since the wooden frame surrounds the painting, we can assume that the dimensions of the frame plus the dimensions of the painting will give us the total dimensions of the picture frame. Makes sense, right?
Let's consider the width of the painting as "x" (in feet). According to the information given, the length of the painting would then be "x + 3" feet.
Now, let's calculate the dimensions of the picture frame, including the wooden frame:
Width of the frame = x + 2 (frame on both sides of the painting)
Length of the frame = (x + 3) + 2 (frame on both sides of the painting)
Area of the frame = Width of frame * Length of frame
But remember, we're given that the area of the picture frame is 286 square feet. So, we set up the equation like this:
(x + 2) * (x + 5) = 286
Now, you can solve the equation to find the value of "x", which represents the width of the painting. Once you have "x", you can calculate the length by adding 3 feet to it.
I hope that helps! Just remember, math + art = the perfect masterpiece!
To solve this problem, let's break it down into steps:
Step 1: Let's assign variables for the dimensions of the painting.
Let's say the width of the painting is "x" feet.
Since the painting is 3 feet longer than it is wide, the length is "x + 3" feet.
So, the dimensions of the painting are x feet (width) and (x + 3) feet (length).
Step 2: Calculate the dimensions of the painting frame.
The wooden frame is 18 inches wide, which is equivalent to 1.5 feet.
The overall width of the painting (including the frame) is x + 3 feet (length of the painting) + 1.5 feet (width of the frame) + 1.5 feet (width of the frame) = x + 6 feet.
The overall length of the painting (including the frame) is x feet (width of the painting) + 1.5 feet (width of the frame) + 1.5 feet (width of the frame) = x + 3 feet.
Step 3: Calculate the area of the picture frame.
The area of a rectangle can be calculated by multiplying its length by its width.
The area of the frame is given as 286 ft^2.
So, (x + 6) feet (overall width of the painting) * (x + 3) feet (overall length of the painting) = 286 ft^2.
Step 4: Solve the equation for x.
Rewrite the equation: (x + 6)(x + 3) = 286.
Expand the equation: x^2 + 9x + 18 = 286.
Move 286 to the left side of the equation: x^2 + 9x + 18 - 286 = 0.
Simplify: x^2 + 9x - 268 = 0.
Step 5: Solve the quadratic equation.
You can solve this equation by factoring, completing the square, or using the quadratic formula. Since the numbers are not easily factorable, we will use the quadratic formula.
The quadratic formula is given by: x = (-b ± √(b^2 - 4ac))/(2a).
For this equation, a = 1, b = 9, and c = -268.
Plug in the values into the quadratic formula and solve for x.
Step 6: Substitute x into the expressions for the dimensions of the painting.
Once you have the values of x, substitute it back into the equation for the dimensions of the painting:
Width = x feet
Length = x + 3 feet
Step 7: Verify the solution.
Double-check if the calculated dimensions satisfy the given conditions of being 3 feet longer and if the frame area is 286 ft^2.
I hope this step-by-step guide helps you solve the problem. Let me know if you have any further questions!
No problem! Let's break down the problem step by step.
1. Determine the area of the painting:
The area of a rectangle can be found by multiplying its length and width. Let's say the width of the painting is 'w' feet. According to the problem, the length of the painting is 3 feet longer than the width, so the length would be 'w + 3' feet. Therefore, the area of the painting would be (w)(w + 3) square feet.
2. Determine the dimensions of the painting including the frame:
To find the dimensions of the painting including the frame, we need to add the width of the frame to both the length and the width of the painting. Since the frame is 18 inches wide, which is equivalent to 1.5 feet, the new width of the painting would be 'w + 1.5' feet, and the new length of the painting would be 'w + 3 + 1.5' feet.
3. Calculate the area of the frame:
The area of the frame is given as 286 square feet. We can find the area of the frame by subtracting the area of the painting from the total area of the frame and painting combined. The area of the frame would be the total area of the frame and painting combined (including the frame) minus the area of the painting alone.
4. Set up the equation and solve for 'w':
The equation for the area of the frame can be written as:
Total area of frame and painting - Area of painting = Area of frame
[(w + 1.5)(w + 3 + 1.5)] - (w)(w + 3) = 286
Now, you can solve this quadratic equation for 'w' using algebraic methods such as factoring, completing the square, or using the quadratic formula. Once you find the value of 'w', you can calculate the length of the painting by adding 3 to 'w', and you'll have the dimensions of the painting.
I hope this helps! Let me know if you have any more questions.