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Tap on the tub. Does it sound like it is solid or hollow? An acrylic tub is light and will sound hollow inside, like you are knocking on a door. A stainless steel tub will make a ticking or clacking sound. A cast iron tub will sound like a deep gong, just as if you hit a cast iron frying pan.
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Look at the bottom of the tub if possible. A cast iron tub will be black, and a steel tub will be metallic, although it may be covered with a layer of paint. An acrylic tub will look like coated plastic.
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Try to lift the tub. If you are removing the tub, it is easy to tell if it is cast iron. A cast iron tub is at least four times heavier than a steel tub.
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Ask your neighbors and the people who owned the house before you. If you live in a housing complex where all the houses are similar, there may be records that state what tubs were installed in the complex. If you are just moving into a house, ask the previous owners who installed the tub and when, and ask them what the tub is made of.