Ask an adult to help peel the cucumbers if desired. Trim the ends, halve lengthwise, and cut crosswise into ¼-inch-thick slices. Peel the carrot and cut into ¼-inch slices.
Put the vegetables in a 1-quart glass jar or other acid-proof container with a tight-fitting lid.
To make the brine, put the vinegar, hot water, sugar, and salt in a heat-proof bowl. Stir the brine until the sugar and salt dissolve. Let it cool for 15 minutes. Then pour it into the jar.
Chill the veggies in the refrigerator for at least three hours.
