Question

the space allowed for the mascot on a schools web page is 112 pixels wide by 70 pixels high. its digital image is 600 pixels wide by 375 pixels high. what is the largest image of the mascot that will fit on the web page? the largest image that will fit on the web page is (square) pixels wide by (square) pixels high. (simplify your answer.)

Explanation:

Step1: Find the scale factor for width

To find the scale factor, we divide the allowed width by the image width. The allowed width is 112 pixels and the image width is 600 pixels. So the scale factor for width is $\frac{112}{600}=\frac{14}{75}\approx0.1867$.

Step2: Find the scale factor for height

The allowed height is 70 pixels and the image height is 375 pixels. So the scale factor for height is $\frac{70}{375}=\frac{14}{75}\approx0.1867$.

Step3: Determine the largest image

Since both scale factors are the same ($\frac{14}{75}$), we can use this scale factor to find the dimensions of the largest image that fits.
For the width: $600\times\frac{14}{75} = 112$ pixels.
For the height: $375\times\frac{14}{75}=70$ pixels. Wait, no, that's the original allowed size. Wait, actually, we need to check which scale factor is more restrictive. But in this case, both scale factors are equal. Wait, no, let's recalculate. Wait, maybe I got it reversed. Wait, the image is 600x375, and the allowed space is 112x70. So we need to find the maximum scale factor $s$ such that $600s\leq112$ and $375s\leq70$. Let's solve for $s$ in both inequalities.

For width: $s\leq\frac{112}{600}=\frac{14}{75}\approx0.1867$.

For height: $s\leq\frac{70}{375}=\frac{14}{75}\approx0.1867$.

So the scale factor is $\frac{14}{75}$. Now, apply this scale factor to the image dimensions.

Width: $600\times\frac{14}{75}= 600\div75\times14 = 8\times14 = 112$ pixels.

Height: $375\times\frac{14}{75}=375\div75\times14 = 5\times14 = 70$ pixels. Wait, but that's the exact size of the allowed space. But let's check if the aspect ratio of the image and the allowed space is the same. The aspect ratio of the image is $\frac{600}{375}=\frac{8}{5}=1.6$. The aspect ratio of the allowed space is $\frac{112}{70}=\frac{8}{5}=1.6$. So they have the same aspect ratio. Therefore, the largest image that fits is the one that scales the image by the scale factor, which results in the allowed dimensions. Wait, but that seems like the image is resized to fit exactly. Let's verify:

If we scale the image (600x375) by $s=\frac{14}{75}$, we get (600(14/75), 375(14/75)) = (112, 70), which is exactly the allowed space. So that's the largest image that fits.

Answer:

112 pixels wide by 70 pixels high.